Solve 5x^2+50x+24=64

$$5 x ^ { 2 } + 50 x + 24 = 64$$

Solve for x (complex solution)

$$x=\sqrt{33}-5\approx 0.744562647$$

$$x=-\left(\sqrt{33}+5\right)\approx -10.744562647$$

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.

$$5x^{2}+50x+24=64$$

Subtract 64 from both sides of the equation.

$$5x^{2}+50x+24-64=64-64$$

Subtracting 64 from itself leaves 0.

$$5x^{2}+50x+24-64=0$$

Subtract 64 from 24.

$$5x^{2}+50x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 50 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\times 5\left(-40\right)}}{2\times 5}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\times 5\left(-40\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-50±\sqrt{2500-20\left(-40\right)}}{2\times 5}$$

Multiply -20 times -40.

$$x=\frac{-50±\sqrt{2500+800}}{2\times 5}$$

Add 2500 to 800.

$$x=\frac{-50±\sqrt{3300}}{2\times 5}$$

Take the square root of 3300.

$$x=\frac{-50±10\sqrt{33}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-50±10\sqrt{33}}{10}$$

Now solve the equation x=\frac{-50±10\sqrt{33}}{10} when ± is plus. Add -50 to 10\sqrt{33}.

$$x=\frac{10\sqrt{33}-50}{10}$$

Divide -50+10\sqrt{33} by 10.

$$x=\sqrt{33}-5$$

Now solve the equation x=\frac{-50±10\sqrt{33}}{10} when ± is minus. Subtract 10\sqrt{33} from -50.

$$x=\frac{-10\sqrt{33}-50}{10}$$

Divide -50-10\sqrt{33} by 10.

$$x=-\sqrt{33}-5$$

The equation is now solved.

$$x=\sqrt{33}-5 x=-\sqrt{33}-5$$

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.

$$5x^{2}+50x+24=64$$

Subtract 24 from both sides of the equation.

$$5x^{2}+50x+24-24=64-24$$

Subtracting 24 from itself leaves 0.

$$5x^{2}+50x=64-24$$

Subtract 24 from 64.

$$5x^{2}+50x=40$$

Divide both sides by 5.

$$\frac{5x^{2}+50x}{5}=\frac{40}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{50}{5}x=\frac{40}{5}$$

Divide 50 by 5.

$$x^{2}+10x=\frac{40}{5}$$

Divide 40 by 5.

$$x^{2}+10x=8$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=8+5^{2}$$

Square 5.

$$x^{2}+10x+25=8+25$$

Add 8 to 25.

$$x^{2}+10x+25=33$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=33$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{33}$$

Simplify.

$$x+5=\sqrt{33} x+5=-\sqrt{33}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{33}-5 x=-\sqrt{33}-5$$

Solve for x

$$x=\sqrt{33}-5\approx 0.744562647$$

$$x=-\sqrt{33}-5\approx -10.744562647$$

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.

$$5x^{2}+50x+24=64$$

Subtract 64 from both sides of the equation.

$$5x^{2}+50x+24-64=64-64$$

Subtracting 64 from itself leaves 0.

$$5x^{2}+50x+24-64=0$$

Subtract 64 from 24.

$$5x^{2}+50x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 50 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\times 5\left(-40\right)}}{2\times 5}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\times 5\left(-40\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-50±\sqrt{2500-20\left(-40\right)}}{2\times 5}$$

Multiply -20 times -40.

$$x=\frac{-50±\sqrt{2500+800}}{2\times 5}$$

Add 2500 to 800.

$$x=\frac{-50±\sqrt{3300}}{2\times 5}$$

Take the square root of 3300.

$$x=\frac{-50±10\sqrt{33}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-50±10\sqrt{33}}{10}$$

Now solve the equation x=\frac{-50±10\sqrt{33}}{10} when ± is plus. Add -50 to 10\sqrt{33}.

$$x=\frac{10\sqrt{33}-50}{10}$$

Divide -50+10\sqrt{33} by 10.

$$x=\sqrt{33}-5$$

Now solve the equation x=\frac{-50±10\sqrt{33}}{10} when ± is minus. Subtract 10\sqrt{33} from -50.

$$x=\frac{-10\sqrt{33}-50}{10}$$

Divide -50-10\sqrt{33} by 10.

$$x=-\sqrt{33}-5$$

The equation is now solved.

$$x=\sqrt{33}-5 x=-\sqrt{33}-5$$

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.

$$5x^{2}+50x+24=64$$

Subtract 24 from both sides of the equation.

$$5x^{2}+50x+24-24=64-24$$

Subtracting 24 from itself leaves 0.

$$5x^{2}+50x=64-24$$

Subtract 24 from 64.

$$5x^{2}+50x=40$$

Divide both sides by 5.

$$\frac{5x^{2}+50x}{5}=\frac{40}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{50}{5}x=\frac{40}{5}$$

Divide 50 by 5.

$$x^{2}+10x=\frac{40}{5}$$

Divide 40 by 5.

$$x^{2}+10x=8$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=8+5^{2}$$

Square 5.

$$x^{2}+10x+25=8+25$$

Add 8 to 25.

$$x^{2}+10x+25=33$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=33$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{33}$$

Simplify.

$$x+5=\sqrt{33} x+5=-\sqrt{33}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{33}-5 x=-\sqrt{33}-5$$