10.2 Solve quadratic equations by completing the square (Page 2/5)

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

$a^{2} - 20 a$

${(a - 10)}^{2}$

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

$b^{2} - 4 b$

${(b - 2)}^{2}$

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Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

$u^{2} - 9 u$

Solution

The coefficient of u is −9.
$\begin{matrix} Find {(\frac{1}{2} b)}^{2} . \\ {(\frac{1}{2} \cdot (− 9))}^{2} \\ {(- \frac{9}{2})}^{2} \\ \frac{81}{4} \end{matrix}$
Add $\frac{81}{4}$ to the binomial to complete the square.	$u^{2} - 9 u + \frac{81}{4}$
Rewrite as a binomial square.	${(u - \frac{9}{2})}^{2}$

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

$m^{2} - 5 m$

${(m - \frac{5}{2})}^{2}$

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

$n^{2} + 13 n$

${(n + \frac{13}{2})}^{2}$

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Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

$p^{2} + \frac{1}{2} p$

Solution

The coefficient of p is $\frac{1}{2}$ .
$\begin{matrix} Find {(\frac{1}{2} b)}^{2} . \\ {(\frac{1}{2} \cdot \frac{1}{2})}^{2} \\ {(\frac{1}{4})}^{2} \\ \frac{1}{16} \end{matrix}$
Add $\frac{1}{16}$ to the binomial to complete the square.	$p^{2} + \frac{1}{2} p + \frac{1}{16}$
Rewrite as a binomial square.	${(p + \frac{1}{4})}^{2}$

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

$p^{2} + \frac{1}{4} p$

${(p + \frac{1}{8})}^{2}$

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Complete the square to make a perfect square trinomial. Write the result as a binomial square.

$q^{2} - \frac{2}{3} q$

${(q - \frac{1}{3})}^{2}$

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Solve quadratic equations of the form x ² + bx + c = 0 by completing the square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square , too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

For example, if we start with the equation $x^{2} + 6 x = 40$ and we want to complete the square on the left, we will add nine to both sides of the equation.

The image shows the equation x squared plus six x equals 40. Below that the equation is rewritten as x squared plus six x plus blank space equals 40 plus blank space. Below that the equation is rewritten again as x squared plus six x plus nine equals 40 plus nine.

Then, we factor on the left and simplify on the right.

{(x + 3)}^{2} = 49

Now the equation is in the form to solve using the Square Root Property. Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.

How to solve a quadratic equation of the form $x^{2} + b x + c = 0$ By completing the square

Solve $x^{2} + 8 x = 48$ by completing the square.

Solution

The image shows the steps to solve the equation x squared plus eight x equals 48. Step one is to isolate the variable terms on one side and the constant terms on the other. The equation already has all the variables on the left.

Step two is to find the quantity half of b squared, the number to complete the square and add it to both sides of the equation. The coefficient of x is eight so b is eight. Take half of eight, which is four and square it to get 16. Add 16 to both sides of the equation to get x squared plus eight x plus 16 equals 48 plus 16.

Step three is to factor the perfect square trinomial as a binomial square. The left side is the perfect square trinomial x squared plus eight x plus 16 which factors to the quantity x plus four squared. Adding on the right side 48 plus 16 is 64. The equation is now the quantity x plus four squared equals 64.

Step four is to use the square root property to make the equation x plus four equals plus or minus the square root of 64.

Step five is to simplify the radical and then solve the two resulting equations. The square root of 64 is eight. The equation can be written as two equations: x plus four equals eight and x plus four equals negative eight. Solving each equation gives x equals four or negative 12.

Step six is to check the solutions. To check the solutions put each answer in the original equation. Substituting x equals four in the original equation to get four squared plus eight times four equals 48. The left side simplifies to 16 plus 32 which is 48. Substituting x equals negative 12 in the original equation to get negative 12 squared plus eight times negative 12 equals 48. The left side simplifies to 144 minus 96 which is 48.

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Solve $c^{2} + 4 c = 5$ by completing the square.

$c = −5, c = 1$

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Solve $d^{2} + 10 d = −9$ by completing the square.

$d = −9, d = −1$

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Solve a quadratic equation of the form $x^{2} + b x + c = 0$ By completing the square.

Isolate the variable terms on one side and the constant terms on the other.
Find ${(\frac{1}{2} \cdot b)}^{2}$ , the number to complete the square. Add it to both sides of the equation.
Factor the perfect square trinomial as a binomial square.
Use the Square Root Property.
Simplify the radical and then solve the two resulting equations.
Check the solutions.

Solve $y^{2} - 6 y = 16$ by completing the square.

Solution

The variable terms are on the left side.
Take half of $- 6$ and square it. $(\frac{1}{2} (− 6))^{2} = 9$
Add 9 to both sides.
Factor the perfect square trinomial as a binomial square.
Use the Square Root Property.
Simplify the radical.
Solve for y .
Rewrite to show two solutions.
Solve the equations.
Check.

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Solve $r^{2} - 4 r = 12$ by completing the square.

$r = −2, r = 6$

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Solve $t^{2} - 10 t = 11$ by completing the square.

$t = −1, t = 11$

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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10.2 Solve quadratic equations by completing the square (Page 2/5)

Solution

Solution

Solve quadratic equations of the form x 2 + bx + c = 0 by completing the square

How to solve a quadratic equation of the form x 2 + b x + c = 0 By completing the square

Solution

Solve a quadratic equation of the form x 2 + b x + c = 0 By completing the square.

Solution

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Solve quadratic equations of the form x ² + bx + c = 0 by completing the square

How to solve a quadratic equation of the form $x^{2} + b x + c = 0$ By completing the square

Solve a quadratic equation of the form $x^{2} + b x + c = 0$ By completing the square.