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Given a quadratic equation with the leading coefficient of 1, factor it.

  1. Find two numbers whose product equals c and whose sum equals b .
  2. Use those numbers to write two factors of the form ( x + k )  or  ( x k ) , where k is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and −2 , the factors are ( x + 1 ) ( x 2 ) .
  3. Solve using the zero-product property by setting each factor equal to zero and solving for the variable.

Solving a quadratic equation by factoring when the leading coefficient is not 1

Factor and solve the equation: x 2 + x 6 = 0.

To factor x 2 + x 6 = 0 , we look for two numbers whose product equals −6 and whose sum equals 1. Begin by looking at the possible factors of −6.

1 ( −6 ) ( −6 ) 1 2 ( −3 ) 3 ( −2 )

The last pair, 3 ( −2 ) sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors.

( x 2 ) ( x + 3 ) = 0

To solve this equation, we use the zero-product property. Set each factor equal to zero and solve.

( x 2 ) ( x + 3 ) = 0 ( x 2 ) = 0 x = 2 ( x + 3 ) = 0 x = −3

The two solutions are 2 and −3. We can see how the solutions relate to the graph in [link] . The solutions are the x- intercepts of y = x 2 + x 6 = 0.

Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 7 to 7. The function x squared plus x minus six equals zero is graphed, with the x-intercepts (-3,0) and (2,0), plotted as well.
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Factor and solve the quadratic equation: x 2 5 x 6 = 0.

( x 6 ) ( x + 1 ) = 0 ; x = 6 , x = 1

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Solve the quadratic equation by factoring

Solve the quadratic equation by factoring: x 2 + 8 x + 15 = 0.

Find two numbers whose product equals 15 and whose sum equals 8. List the factors of 15.

1 15 3 5 ( −1 ) ( −15 ) ( −3 ) ( −5 )

The numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve.

( x + 3 ) ( x + 5 ) = 0 ( x + 3 ) = 0 x = −3 ( x + 5 ) = 0 x = −5

The solutions are −3 and −5.

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Solve the quadratic equation by factoring: x 2 4 x 21 = 0.

( x −7 ) ( x + 3 ) = 0 , x = 7 , x = −3.

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Using the zero-product property to solve a quadratic equation written as the difference of squares

Solve the difference of squares equation using the zero-product property: x 2 9 = 0.

Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property.

x 2 9 = 0 ( x 3 ) ( x + 3 ) = 0 ( x 3 ) = 0 x = 3 ( x + 3 ) = 0 x = −3

The solutions are 3 and −3.

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Solve by factoring: x 2 25 = 0.

( x + 5 ) ( x −5 ) = 0 , x = −5 , x = 5.

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Solving a quadratic equation by factoring when the leading coefficient is not 1

When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let’s review the grouping procedures:

  1. With the quadratic in standard form, a x 2 + b x + c = 0 , multiply a c .
  2. Find two numbers whose product equals a c and whose sum equals b .
  3. Rewrite the equation replacing the b x term with two terms using the numbers found in step 1 as coefficients of x.
  4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
  5. Factor out the expression in parentheses.
  6. Set the expressions equal to zero and solve for the variable.

Questions & Answers

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The numbers cannot be combined with the x
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2x + 7 =19
humberto
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because x is 6
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what is the value of x in 4x-2+3
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give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
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if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
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4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
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A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
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A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
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Practice Key Terms 7

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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