Solve 2x(2x+10)=5x+2(x-1)

$$2 x ( 2 x + 10 ) = 5 x + 2 ( x - 1 )$$

Solve for x

$$x=\frac{\sqrt{137}-13}{8}\approx -0.161912511$$

$$x=\frac{-\sqrt{137}-13}{8}\approx -3.088087489$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 2x+10.

$$4x^{2}+20x=5x+2\left(x-1\right)$$

Use the distributive property to multiply 2 by x-1.

$$4x^{2}+20x=5x+2x-2$$

Combine 5x and 2x to get 7x.

$$4x^{2}+20x=7x-2$$

Subtract 7x from both sides.

$$4x^{2}+20x-7x=-2$$

Combine 20x and -7x to get 13x.

$$4x^{2}+13x=-2$$

Add 2 to both sides.

$$4x^{2}+13x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 13 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 4\times 2}}{2\times 4}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 4\times 2}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-13±\sqrt{169-16\times 2}}{2\times 4}$$

Multiply -16 times 2.

$$x=\frac{-13±\sqrt{169-32}}{2\times 4}$$

Add 169 to -32.

$$x=\frac{-13±\sqrt{137}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-13±\sqrt{137}}{8}$$

Now solve the equation x=\frac{-13±\sqrt{137}}{8} when ± is plus. Add -13 to \sqrt{137}.

$$x=\frac{\sqrt{137}-13}{8}$$

Now solve the equation x=\frac{-13±\sqrt{137}}{8} when ± is minus. Subtract \sqrt{137} from -13.

$$x=\frac{-\sqrt{137}-13}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{137}-13}{8} x=\frac{-\sqrt{137}-13}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 2x+10.

$$4x^{2}+20x=5x+2\left(x-1\right)$$

Use the distributive property to multiply 2 by x-1.

$$4x^{2}+20x=5x+2x-2$$

Combine 5x and 2x to get 7x.

$$4x^{2}+20x=7x-2$$

Subtract 7x from both sides.

$$4x^{2}+20x-7x=-2$$

Combine 20x and -7x to get 13x.

$$4x^{2}+13x=-2$$

Divide both sides by 4.

$$\frac{4x^{2}+13x}{4}=-\frac{2}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{13}{4}x=-\frac{2}{4}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{13}{4}x=-\frac{1}{2}$$

Divide \frac{13}{4}, the coefficient of the x term, by 2 to get \frac{13}{8}. Then add the square of \frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{4}x+\left(\frac{13}{8}\right)^{2}=-\frac{1}{2}+\left(\frac{13}{8}\right)^{2}$$

Square \frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=-\frac{1}{2}+\frac{169}{64}$$

Add -\frac{1}{2} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=\frac{137}{64}$$

Factor x^{2}+\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{8}\right)^{2}=\frac{137}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{8}\right)^{2}}=\sqrt{\frac{137}{64}}$$

Simplify.

$$x+\frac{13}{8}=\frac{\sqrt{137}}{8} x+\frac{13}{8}=-\frac{\sqrt{137}}{8}$$

Subtract \frac{13}{8} from both sides of the equation.

$$x=\frac{\sqrt{137}-13}{8} x=\frac{-\sqrt{137}-13}{8}$$