Solve x^2+7x-8=5x^2-1

$${ x }^{ 2 } +7x-8=5 { x }^{ 2 } -1$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{7}i+7}{8}\approx 0.875-0.992156742i$$

$$x=\frac{7+3\sqrt{7}i}{8}\approx 0.875+0.992156742i$$

Steps Using the Quadratic Formula

Subtract 5x^{2} from both sides.

$$x^{2}+7x-8-5x^{2}=-1$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+7x-8=-1$$

Add 1 to both sides.

$$-4x^{2}+7x-8+1=0$$

Add -8 and 1 to get -7.

$$-4x^{2}+7x-7=0$$

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

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

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-4\right)\left(-7\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-7±\sqrt{49+16\left(-7\right)}}{2\left(-4\right)}$$

Multiply 16 times -7.

$$x=\frac{-7±\sqrt{49-112}}{2\left(-4\right)}$$

Add 49 to -112.

$$x=\frac{-7±\sqrt{-63}}{2\left(-4\right)}$$

Take the square root of -63.

$$x=\frac{-7±3\sqrt{7}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-7±3\sqrt{7}i}{-8}$$

Now solve the equation x=\frac{-7±3\sqrt{7}i}{-8} when ± is plus. Add -7 to 3i\sqrt{7}.

$$x=\frac{-7+3\sqrt{7}i}{-8}$$

Divide -7+3i\sqrt{7} by -8.

$$x=\frac{-3\sqrt{7}i+7}{8}$$

Now solve the equation x=\frac{-7±3\sqrt{7}i}{-8} when ± is minus. Subtract 3i\sqrt{7} from -7.

$$x=\frac{-3\sqrt{7}i-7}{-8}$$

Divide -7-3i\sqrt{7} by -8.

$$x=\frac{7+3\sqrt{7}i}{8}$$

The equation is now solved.

$$x=\frac{-3\sqrt{7}i+7}{8} x=\frac{7+3\sqrt{7}i}{8}$$

Steps for Completing the Square

Subtract 5x^{2} from both sides.

$$x^{2}+7x-8-5x^{2}=-1$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+7x-8=-1$$

Add 8 to both sides.

$$-4x^{2}+7x=-1+8$$

Add -1 and 8 to get 7.

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

Divide both sides by -4.

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

Dividing by -4 undoes the multiplication by -4.

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

Divide 7 by -4.

$$x^{2}-\frac{7}{4}x=\frac{7}{-4}$$

Divide 7 by -4.

$$x^{2}-\frac{7}{4}x=-\frac{7}{4}$$

Divide -\frac{7}{4}, the coefficient of the x term, by 2 to get -\frac{7}{8}. Then add the square of -\frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=-\frac{7}{4}+\frac{49}{64}$$

Add -\frac{7}{4} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=-\frac{63}{64}$$

Factor x^{2}-\frac{7}{4}x+\frac{49}{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}.

$$\left(x-\frac{7}{8}\right)^{2}=-\frac{63}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{-\frac{63}{64}}$$

Simplify.

$$x-\frac{7}{8}=\frac{3\sqrt{7}i}{8} x-\frac{7}{8}=-\frac{3\sqrt{7}i}{8}$$

Add \frac{7}{8} to both sides of the equation.

$$x=\frac{7+3\sqrt{7}i}{8} x=\frac{-3\sqrt{7}i+7}{8}$$