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Solving a quadratic equation using grouping

Use grouping to factor and solve the quadratic equation: 4 x 2 + 15 x + 9 = 0.

First, multiply a c : 4 ( 9 ) = 36. Then list the factors of 36.

1 36 2 18 3 12 4 9 6 6

The only pair of factors that sums to 15 is 3 + 12. Rewrite the equation replacing the b term, 15 x , with two terms using 3 and 12 as coefficients of x . Factor the first two terms, and then factor the last two terms.

4 x 2 + 3 x + 12 x + 9 = 0 x ( 4 x + 3 ) + 3 ( 4 x + 3 ) = 0 ( 4 x + 3 ) ( x + 3 ) = 0

Solve using the zero-product property.

( 4 x + 3 ) ( x + 3 ) = 0 ( 4 x + 3 ) = 0 x = 3 4 ( x + 3 ) = 0 x = 3

The solutions are 3 4 , and −3. See [link] .

Coordinate plane with the x-axis ranging from negative 6 to 2 with every other tick mark labeled and the y-axis ranging from negative 6 to 2 with each tick mark numbered. The equation: four x squared plus fifteen x plus nine is graphed with its x-intercepts: (-3/4,0) and (-3,0) plotted as well.
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Solve using factoring by grouping: 12 x 2 + 11 x + 2 = 0.

( 3 x + 2 ) ( 4 x + 1 ) = 0 , x = 2 3 , x = 1 4

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Solving a polynomial of higher degree by factoring

Solve the equation by factoring: −3 x 3 5 x 2 2 x = 0.

This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out x from all of the terms and then proceed with grouping.

−3 x 3 5 x 2 2 x = 0 x ( 3 x 2 + 5 x + 2 ) = 0

Use grouping on the expression in parentheses.

x ( 3 x 2 + 3 x + 2 x + 2 ) = 0 x [ 3 x ( x + 1 ) + 2 ( x + 1 ) ] = 0 x ( 3 x + 2 ) ( x + 1 ) = 0

Now, we use the zero-product property. Notice that we have three factors.

x = 0 x = 0 3 x + 2 = 0 x = 2 3 x + 1 = 0 x = −1

The solutions are 0 , 2 3 , and −1.

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Solve by factoring: x 3 + 11 x 2 + 10 x = 0.

x = 0 , x = −10 , x = −1

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Using the square root property

When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property    , in which we isolate the x 2 term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the x 2 term so that the square root property can be used.

The square root property

With the x 2 term isolated, the square root property states that:

if x 2 = k , then x = ± k

where k is a nonzero real number.

Given a quadratic equation with an x 2 term but no x term, use the square root property to solve it.

  1. Isolate the x 2 term on one side of the equal sign.
  2. Take the square root of both sides of the equation, putting a ± sign before the expression on the side opposite the squared term.
  3. Simplify the numbers on the side with the ± sign.

Solving a simple quadratic equation using the square root property

Solve the quadratic using the square root property: x 2 = 8.

Take the square root of both sides, and then simplify the radical. Remember to use a ± sign before the radical symbol.

x 2 = 8 x = ± 8 = ± 2 2

The solutions are 2 2 , −2 2 .

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Solving a quadratic equation using the square root property

Solve the quadratic equation: 4 x 2 + 1 = 7.

First, isolate the x 2 term. Then take the square root of both sides.

4 x 2 + 1 = 7 4 x 2 = 6 x 2 = 6 4 x = ± 6 2

The solutions are 6 2 , and 6 2 .

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Solve the quadratic equation using the square root property: 3 ( x 4 ) 2 = 15.

x = 4 ± 5

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Completing the square

Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation    known as completing the square    . Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a , must equal 1. If it does not, then divide the entire equation by a . Then, we can use the following procedures to solve a quadratic equation by completing the square.

Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
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
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
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?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb
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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