Solve 5x+5=4x^2

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

Solve for x

$$x = \frac{\sqrt{105} + 5}{8} \approx 1.905868846$$

$$x=\frac{5-\sqrt{105}}{8}\approx -0.655868846$$

Steps Using the Quadratic Formula

Subtract 4x^{2} from both sides.

$$5x+5-4x^{2}=0$$

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.

$$-4x^{2}+5x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 5 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-4\right)\times 5}}{2\left(-4\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-4\right)\times 5}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-5±\sqrt{25+16\times 5}}{2\left(-4\right)}$$

Multiply 16 times 5.

$$x=\frac{-5±\sqrt{25+80}}{2\left(-4\right)}$$

Add 25 to 80.

$$x=\frac{-5±\sqrt{105}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-5±\sqrt{105}}{-8}$$

Now solve the equation x=\frac{-5±\sqrt{105}}{-8} when ± is plus. Add -5 to \sqrt{105}.

$$x=\frac{\sqrt{105}-5}{-8}$$

Divide -5+\sqrt{105} by -8.

$$x=\frac{5-\sqrt{105}}{8}$$

Now solve the equation x=\frac{-5±\sqrt{105}}{-8} when ± is minus. Subtract \sqrt{105} from -5.

$$x=\frac{-\sqrt{105}-5}{-8}$$

Divide -5-\sqrt{105} by -8.

$$x=\frac{\sqrt{105}+5}{8}$$

The equation is now solved.

$$x=\frac{5-\sqrt{105}}{8} x=\frac{\sqrt{105}+5}{8}$$

Steps for Completing the Square

Subtract 4x^{2} from both sides.

$$5x+5-4x^{2}=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -4.

$$\frac{-4x^{2}+5x}{-4}=-\frac{5}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{5}{-4}x=-\frac{5}{-4}$$

Divide 5 by -4.

$$x^{2}-\frac{5}{4}x=-\frac{5}{-4}$$

Divide -5 by -4.

$$x^{2}-\frac{5}{4}x=\frac{5}{4}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=\frac{5}{4}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{5}{4}+\frac{25}{64}$$

Add \frac{5}{4} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{105}{64}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{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}.

$$\left(x-\frac{5}{8}\right)^{2}=\frac{105}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{\frac{105}{64}}$$

Simplify.

$$x-\frac{5}{8}=\frac{\sqrt{105}}{8} x-\frac{5}{8}=-\frac{\sqrt{105}}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$x=\frac{\sqrt{105}+5}{8} x=\frac{5-\sqrt{105}}{8}$$