Solve 2x^2-38x+176=0

$$2 { x }^{ 2 } -38x+176 = 0$$

Solve for x

$$x=8$$

$$x=11$$

Steps Using Factoring By Grouping

Divide both sides by 2.

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

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

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.

$$2x^{2}-38x+176=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -38 for b, and 176 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\times 2\times 176}}{2\times 2}$$

Square -38.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\times 2\times 176}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-38\right)±\sqrt{1444-8\times 176}}{2\times 2}$$

Multiply -8 times 176.

$$x=\frac{-\left(-38\right)±\sqrt{1444-1408}}{2\times 2}$$

Add 1444 to -1408.

$$x=\frac{-\left(-38\right)±\sqrt{36}}{2\times 2}$$

Take the square root of 36.

$$x=\frac{-\left(-38\right)±6}{2\times 2}$$

The opposite of -38 is 38.

$$x=\frac{38±6}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{38±6}{4}$$

Now solve the equation x=\frac{38±6}{4} when ± is plus. Add 38 to 6.

$$x=\frac{44}{4}$$

Divide 44 by 4.

$$x=11$$

Now solve the equation x=\frac{38±6}{4} when ± is minus. Subtract 6 from 38.

$$x=\frac{32}{4}$$

Divide 32 by 4.

$$x=8$$

The equation is now solved.

$$x=11 x=8$$

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.

$$2x^{2}-38x+176=0$$

Subtract 176 from both sides of the equation.

$$2x^{2}-38x+176-176=-176$$

Subtracting 176 from itself leaves 0.

$$2x^{2}-38x=-176$$

Divide both sides by 2.

$$\frac{2x^{2}-38x}{2}=-\frac{176}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{38}{2}\right)x=-\frac{176}{2}$$

Divide -38 by 2.

$$x^{2}-19x=-\frac{176}{2}$$

Divide -176 by 2.

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

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.

$$x=11 x=8$$