Solve 7x+11x+8x^2-7=0

$$7 x + 11 x + 8 x ^ { 2 } - 7 = 0$$

Solve for x

$$x=\frac{\sqrt{137}-9}{8}\approx 0.338087489$$

$$x=\frac{-\sqrt{137}-9}{8}\approx -2.588087489$$

Steps Using the Quadratic Formula

Combine 7x and 11x to get 18x.

$$18x+8x^{2}-7=0$$

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.

$$8x^{2}+18x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 18 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 8\left(-7\right)}}{2\times 8}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 8\left(-7\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-18±\sqrt{324-32\left(-7\right)}}{2\times 8}$$

Multiply -32 times -7.

$$x=\frac{-18±\sqrt{324+224}}{2\times 8}$$

Add 324 to 224.

$$x=\frac{-18±\sqrt{548}}{2\times 8}$$

Take the square root of 548.

$$x=\frac{-18±2\sqrt{137}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-18±2\sqrt{137}}{16}$$

Now solve the equation x=\frac{-18±2\sqrt{137}}{16} when ± is plus. Add -18 to 2\sqrt{137}.

$$x=\frac{2\sqrt{137}-18}{16}$$

Divide -18+2\sqrt{137} by 16.

$$x=\frac{\sqrt{137}-9}{8}$$

Now solve the equation x=\frac{-18±2\sqrt{137}}{16} when ± is minus. Subtract 2\sqrt{137} from -18.

$$x=\frac{-2\sqrt{137}-18}{16}$$

Divide -18-2\sqrt{137} by 16.

$$x=\frac{-\sqrt{137}-9}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{137}-9}{8} x=\frac{-\sqrt{137}-9}{8}$$

Steps for Completing the Square

Combine 7x and 11x to get 18x.

$$18x+8x^{2}-7=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$18x+8x^{2}=7$$

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.

$$8x^{2}+18x=7$$

Divide both sides by 8.

$$\frac{8x^{2}+18x}{8}=\frac{7}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{18}{8}x=\frac{7}{8}$$

Reduce the fraction \frac{18}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{9}{4}x=\frac{7}{8}$$

Divide \frac{9}{4}, the coefficient of the x term, by 2 to get \frac{9}{8}. Then add the square of \frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{4}x+\left(\frac{9}{8}\right)^{2}=\frac{7}{8}+\left(\frac{9}{8}\right)^{2}$$

Square \frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{7}{8}+\frac{81}{64}$$

Add \frac{7}{8} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{137}{64}$$

Factor x^{2}+\frac{9}{4}x+\frac{81}{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}.

$$\left(x+\frac{9}{8}\right)^{2}=\frac{137}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{8}\right)^{2}}=\sqrt{\frac{137}{64}}$$

Simplify.

$$x+\frac{9}{8}=\frac{\sqrt{137}}{8} x+\frac{9}{8}=-\frac{\sqrt{137}}{8}$$

Subtract \frac{9}{8} from both sides of the equation.

$$x=\frac{\sqrt{137}-9}{8} x=\frac{-\sqrt{137}-9}{8}$$