Solve x^2+11x+63=35

$${ x }^{ 2 } +11x+63 = 35$$

Solve for x

$$x=-7$$

$$x=-4$$

Steps Using Factoring

Subtract 35 from both sides.

$$x^{2}+11x+63-35=0$$

Subtract 35 from 63 to get 28.

$$x^{2}+11x+28=0$$

To solve the equation, factor x^{2}+11x+28 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=11 ab=28$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=4 b=7$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x+4\right)\left(x+7\right)$$

To find equation solutions, solve x+4=0 and x+7=0.

$$x=-4 x=-7$$

Steps Using Factoring By Grouping

Subtract 35 from both sides.

$$x^{2}+11x+63-35=0$$

Subtract 35 from 63 to get 28.

$$x^{2}+11x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 28=28$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=4 b=7$$

Rewrite x^{2}+11x+28 as \left(x^{2}+4x\right)+\left(7x+28\right).

$$\left(x^{2}+4x\right)+\left(7x+28\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x+4\right)+7\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+7\right)$$

To find equation solutions, solve x+4=0 and x+7=0.

$$x=-4 x=-7$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+11x+63=35$$

Subtract 35 from both sides of the equation.

$$x^{2}+11x+63-35=35-35$$

Subtracting 35 from itself leaves 0.

$$x^{2}+11x+63-35=0$$

Subtract 35 from 63.

$$x^{2}+11x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 28}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 28}}{2}$$

Multiply -4 times 28.

$$x=\frac{-11±\sqrt{121-112}}{2}$$

Add 121 to -112.

$$x=\frac{-11±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-11±3}{2}$$

Now solve the equation x=\frac{-11±3}{2} when ± is plus. Add -11 to 3.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

Now solve the equation x=\frac{-11±3}{2} when ± is minus. Subtract 3 from -11.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=-4 x=-7$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+11x+63=35$$

Subtract 63 from both sides of the equation.

$$x^{2}+11x+63-63=35-63$$

Subtracting 63 from itself leaves 0.

$$x^{2}+11x=35-63$$

Subtract 63 from 35.

$$x^{2}+11x=-28$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-28+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-28+\frac{121}{4}$$

Add -28 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{9}{4}$$

Factor x^{2}+11x+\frac{121}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{3}{2} x+\frac{11}{2}=-\frac{3}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-4 x=-7$$