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Solving quadratic equations | Lesson

What are quadratic equations, and how frequently do they appear on the test?

A quadratic equation is an equation with a
variable to the second power
as its highest power term. For example, in the quadratic equation 3, x, squared, minus, 5, x, minus, 2, equals, 0:
  • x is the variable, which represents a number whose value we don't know yet.
  • The squared is the power or exponent. An exponent of 2 means the variable is
    multiplied by itself
    .
  • 3 and minus, 5 are the coefficients, or constant multiples of x, squared and x. 3, x, squared is a single
    term
    , as is minus, 5, x.
  • minus, 2 is a constant term.
In this lesson, we'll learn to:
  1. Solve quadratic equations in several different ways
  2. Determine the number of solutions to a quadratic equation without solving
On your official SAT, you'll likely see 0 to 2 questions that test your ability to solve quadratic equations—more when you include quadratic word problems, linear and quadratic systems, and graphing quadratic functions.
You can learn anything. Let's do this!

How do I solve quadratic equations using square roots?

Solving quadratics by taking square roots

Khan Academy video wrapper
Solving quadratics by taking square rootsSee video transcript

When can I solve by taking square roots?

Quadratic equations without x-terms such as 2, x, squared, equals, 32 can be solved without setting a quadratic expression equal to 0. Instead, we can isolate x, squared and use the square root operation to solve for x.
When solving quadratic equations by taking square roots, both the positive and negative square roots are solutions to the equation. This is because when we square a solution, the result is always positive.
For example, for the equation x, squared, equals, 4, both 2 and minus, 2 are solutions:
  • 2, squared, equals, start superscript, \checkmark, end superscript, 4
  • left parenthesis, minus, 2, right parenthesis, squared, equals, start superscript, \checkmark, end superscript, 4
When solving quadratic equations without x-terms:
  1. Isolate x, squared.
  2. Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

Example: What values of x satisfy the equation 2, x, squared, equals, 18 ?

Try it!

TRY: identify the steps to solving a quadratic equation
x, squared, minus, 3, equals, 13
We can solve the quadratic equation above by first
both sides of the equation, which gives us the equation x, squared, equals, 16.
Next, we can take the square root of both sides of the equation, which gives us the solution(s)
.


What is the zero product property, and how do I use it to solve quadratic equations?

Zero product property

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Zero product propertySee video transcript

Zero product property and factored quadratic equations

The zero product property states that if a, b, equals, 0, then either a or b is equal to 0.
The zero product property lets us solve factored quadratic equations by solving two linear equations. For a quadratic equation such as left parenthesis, x, minus, 5, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, equals, 0, we know that either x, minus, 5, equals, 0 or x, plus, 2, equals, 0. Solving these two linear equations gives us the two solutions to the quadratic equation.
To solve a factored quadratic equation using the zero product property:
  1. Set each factor equal to 0.
  2. Solve the equations from Step 1. The solutions to the linear equations are also solutions to the quadratic equation.

Example: What are the solutions to the equation left parenthesis, x, minus, 4, right parenthesis, left parenthesis, 3, x, plus, 1, right parenthesis, equals, 0 ?

Try it!

TRY: use factors to determine the solutions
2, x, squared, plus, x, minus, 3, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, 2, x, plus, 3, right parenthesis
The equation above shows the factors of the quadratic expression 2, x, squared, plus, x, minus, 3. Which of the following equations, when solved, give us the solutions to the equation 2, x, squared, plus, x, minus, 3, equals, 0 ?
Choose all answers that apply:
Choose all answers that apply:


How do I solve quadratic equations by factoring?

Solving quadratics by factoring

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Solving quadratics by factoringSee video transcript

Solving factorable quadratic equations

If we can write a quadratic expression as the product of two linear expressions (factors), then we can use those linear expressions to calculate the solutions to the quadratic equation.
In this lesson, we'll focus on factorable quadratic equations with 1 as the coefficient of the x, squared term, such as x, squared, minus, 2, x, minus, 3, equals, 0. For more advanced factoring techniques, including special factoring and factoring quadratic expressions with x, squared coefficients other than 1, check out the Structure in expressions lesson.
Recognizing factors of quadratic expressions takes practice. The factors will be in the form left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis, where a and b fulfill the following criteria:
  • The sum of a and b is equal to the coefficient of the x-term in the unfactored quadratic expression.
  • The product of a and b is equal to the constant term of the unfactored quadratic expression.
For example, we can solve the equation x, squared, minus, 2, x, minus, 3, equals, 0 by factoring x, squared, minus, 2, x, minus, 3 into left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis, where:
  • a, plus, b is equal to the coefficient of the x-term, minus, 2.
  • a, b is equal to the constant term, minus, 3.
minus, 3 and 1 would work:
  • minus, 3, plus, 1, equals, minus, 2
  • left parenthesis, minus, 3, right parenthesis, left parenthesis, 1, right parenthesis, equals, minus, 3
This means we can rewrite x, squared, minus, 2, x, minus, 3, equals, 0 as left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, equals, 0 and solve the quadratic equation using the zero product property. Keep mind that a and b are not themselves solutions to the quadratic equation!
When solving factorable quadratic equations in the form x, squared, plus, b, x, plus, c, equals, 0:
  1. Rewrite the quadratic expression as the product of two factors. The two factors are linear expressions with an x-term and a constant term. The sum of the constant terms is equal to b, and the product of the constant terms is equal to c.
  2. Set each factor equal to 0.
  3. Solve the equations from Step 2. The solutions to the linear equations are also solutions to the quadratic equation.

Example: What are the solutions to the equation x, squared, plus, 4, x, minus, 5, equals, 0 ?

Try it!

Try: match the equivalent quadratic expressions
Match each factored expression to its equivalent unfactored expression in the table below.


How do I use the quadratic formula?

The quadratic formula

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The quadratic formulaSee video transcript

Using the quadratic formula to solve equations and determine the number of solutions

Not all quadratic expressions are factorable, and not all factorable quadratic expressions are easy to factor. The quadratic formula gives us a way to solve any quadratic equation as long as we can plug the correct values into the formula and evaluate.
For start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170, equals, 0:
x, equals, start fraction, minus, start color #ca337c, b, end color #ca337c, plus minus, square root of, start color #ca337c, b, end color #ca337c, squared, minus, 4, start color #7854ab, a, end color #7854ab, start color #208170, c, end color #208170, end square root, divided by, 2, start color #7854ab, a, end color #7854ab, end fraction
Note: the quadratic formula is not provided in the reference section of the SAT! You'll have to memorize the formula to use it.

What are the steps?

To solve a quadratic equation using the quadratic formula:
  1. Rewrite the equation in the form a, x, squared, plus, b, x, plus, c, equals, 0.
  2. Substitute the values of a, b, and c into the quadratic formula, shown below.
x, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction
  1. Evaluate x.

Example: What are the solutions to the equation x, squared, minus, 6, x, equals, 9 ?

The b, squared, minus, 4, a, c portion of the quadratic formula is called the discriminant. The value of b, minus, 4, a, c tells us the number of unique real solutions the equation has:
  • If b, squared, minus, 4, a, c, is greater than, 0, then square root of, b, squared, minus, 4, a, c, end square root is a real number, and the quadratic equation has two real solutions, start fraction, minus, b, minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction and start fraction, minus, b, plus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction.
  • If b, squared, minus, 4, a, c, equals, 0, then square root of, b, squared, minus, 4, a, c, end square root is also 0, and the quadratic formula simplifies to start fraction, minus, b, divided by, 2, a, end fraction, which means the quadratic equation has one real solution.
  • If b, squared, minus, 4, a, c, is less than, 0, then square root of, b, squared, minus, 4, a, c, end square root is an imaginary number, which means the quadratic equation has no real solutions.

Try it!

Try: set up for the quadratic formula
7, x, squared, plus, 6, x, minus, 1, equals, 0
If we want to use the quadratic formula to solve the equation above, what are the values of a, b, and c we should plug into the quadratic formula?
a, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
b, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
c, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Try: use the discriminant to find the number of solutions
7, x, squared, plus, 6, x, minus, 1, equals, 0
The value of the
discriminant
for the quadratic equation above is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.
Because the discriminant is
, the equation has
.


Try: substitute into the quadratic formula
7, x, squared, plus, 6, x, minus, 1, equals, 0
Which of the following expressions, when evaluated, gives the solutions to the equation above?
Choose 1 answer:
Choose 1 answer:


Your turn!

Practice: Solve quadratic equation using square root
If start fraction, 1, divided by, 2, end fraction, x, squared, equals, 32 and x, is greater than, 0, what is the value of x ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Practice: Solve quadratic equation by factoring
x, squared, plus, x, minus, 56, equals, 0
What are the solutions to the equation above?
Choose 1 answer:
Choose 1 answer:


Practice: Solve quadratic equation with the quadratic formula
Which of the following values of x satisfy the equation minus, 3, x, squared, plus, 12, x, plus, 4, equals, 0 ?
Choose 1 answer:
Choose 1 answer:


Practice: Determine the condition for one real solution
If a, x, squared, plus, 8, x, plus, 1, equals, 0, for what value of a does the equation have exactly one real solution?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Things to remember

For a, x, squared, plus, b, x, plus, c, equals, 0:
x, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction
  • If b, squared, minus, 4, a, c, is greater than, 0, then the equation has 2 unique real solutions.
  • If b, squared, minus, 4, a, c, equals, 0, then the equation has 1 unique real solution.
  • If b, squared, minus, 4, a, c, is less than, 0, then the equation has no real solution.

Want to join the conversation?

  • blobby green style avatar for user oluwademiladeniyi
    What's a quick way to know if you should use quadratic formula or factorization method to solve a quadratic equation
    (3 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      It largely depends on how good you are at factoring. Factoring is generally faster, but if it's a challenge to try and find the terms then the quadratic formula is more consistent. If when looking for factors you find yourself stuck for maybe more than 30 seconds, you might want to move onto the quadratic method, which you know will work every time. If there is no coefficient on the x^2 term (like in x^2 + 6x + 9), factoring is usually easier. Quadratics with "weird" coefficients (like radicals or decimals) might be better on your calculator in the quadratic formula than factoring by hand. It's really a case by case basis on which to choose, and as you practice more you'll develop that muscle.
      (3 votes)
  • blobby green style avatar for user Maryam Afshar
    I think the answer to this problem is incorrect
    x2+x−56=0x, squared, plus, x, minus, 56, equals, 0
    What are the solutions to the equation above?
    (0 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user abizar05
    -3x^2 + 12x + 4 = 0
    This is the second last question, for this equation, for the answer given in in the explanation, why doesn't the four get square rooted as well when the 8 is divided in to 2*4
    (2 votes)
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  • blobby green style avatar for user khanghanerao
    What are the dots for in the square root?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user ibrahm.qurneh2000
    how dose x=5 = to 0

    and how dose 2/5 =0
    (1 vote)
    Default Khan Academy avatar avatar for user
    • boggle green style avatar for user Valentine Hartman
      If you're talking about the zero method, then I'm not 100% sure, but I think you just need to write it like x=-5 and x=-5/5. You need to add it to both sides of the equation (x+5=0 5-5=0 0-5=-5 x=-5). The whole part where you need to figure out which part makes it zero is just if you want to thoroughly check it.
      (1 vote)
  • spunky sam blue style avatar for user Dami.D
    at sal uses the numerator(2) to divide the denominator(-6), so how did -12 also reduce to -6 because it is really confusing me
    (1 vote)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Hecretary Bird
      This is the distributive property of multiplication/division. Just like how (6x + 12y)/6 means that we need to divide both terms by 2 and end up with x + 2y, the same goes for the +/- in the quadratic formula.
      Sal doesn't divide by 6, he instead divides by 2 in order to cancel out the coefficient on the square root. This means that the denominator becomes -3, and the -12 becomes -6.
      (1 vote)
  • male robot hal style avatar for user Vedangah
    in the zero product property solutions, what do you do if both are simultaneously equal to zero
    for example in (2x-1)(x+4)=0 --- what happens when both (2x+1) and (x+4) are zero at the same time? (since 0 x 0 will still be equal to 0)
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user lili3020santos
    “Which of the following values of x satisfy the equation -3x^2+12x+4=0?”

    how the answer for this question has a positive denominator?
    Cause in the whole equation he is negative, i didnt understand how it got positive
    (1 vote)
    Default Khan Academy avatar avatar for user