Solve x^2=19x-88

$$x ^ { 2 } = 19 x - 88$$

Solve for x

$$x=8$$

$$x=11$$

Steps Using Factoring

Subtract 19x from both sides.

$$x^{2}-19x=-88$$

Add 88 to both sides.

$$x^{2}-19x+88=0$$

To solve the equation, factor x^{2}-19x+88 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.

$$a+b=-19 ab=88$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 88.

$$-1,-88 -2,-44 -4,-22 -8,-11$$

Calculate the sum for each pair.

$$-1-88=-89 -2-44=-46 -4-22=-26 -8-11=-19$$

The solution is the pair that gives sum -19.

$$a=-11 b=-8$$

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

$$\left(x-11\right)\left(x-8\right)$$

To find equation solutions, solve x-11=0 and x-8=0.

$$x=11 x=8$$

Steps Using Factoring By Grouping

Subtract 19x from both sides.

$$x^{2}-19x=-88$$

Add 88 to both sides.

$$x^{2}-19x+88=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+88. To find a and b, set up a system to be solved.

$$a+b=-19 ab=1\times 88=88$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 88.

$$-1,-88 -2,-44 -4,-22 -8,-11$$

Calculate the sum for each pair.

$$-1-88=-89 -2-44=-46 -4-22=-26 -8-11=-19$$

The solution is the pair that gives sum -19.

$$a=-11 b=-8$$

Rewrite x^{2}-19x+88 as \left(x^{2}-11x\right)+\left(-8x+88\right).

$$\left(x^{2}-11x\right)+\left(-8x+88\right)$$

Factor out x in the first and -8 in the second group.

$$x\left(x-11\right)-8\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x-8\right)$$

To find equation solutions, solve x-11=0 and x-8=0.

$$x=11 x=8$$

Steps Using the Quadratic Formula

Subtract 19x from both sides.

$$x^{2}-19x=-88$$

Add 88 to both sides.

$$x^{2}-19x+88=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -19 for b, and 88 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 88}}{2}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 88}}{2}$$

Multiply -4 times 88.

$$x=\frac{-\left(-19\right)±\sqrt{361-352}}{2}$$

Add 361 to -352.

$$x=\frac{-\left(-19\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-\left(-19\right)±3}{2}$$

The opposite of -19 is 19.

$$x=\frac{19±3}{2}$$

Now solve the equation x=\frac{19±3}{2} when ± is plus. Add 19 to 3.

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

Now solve the equation x=\frac{19±3}{2} when ± is minus. Subtract 3 from 19.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

The equation is now solved.

$$x=11 x=8$$

Steps for Completing the Square

Subtract 19x from both sides.

$$x^{2}-19x=-88$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-88+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=-88+\frac{361}{4}$$

Add -88 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=\frac{9}{4}$$

Factor x^{2}-19x+\frac{361}{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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{3}{2} x-\frac{19}{2}=-\frac{3}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=11 x=8$$