Solve 23.15=24.6x-0.5*9.8x^2

$$23.15=24.6x-0.5 \times 9.8 { x }^{ 2 }$$

Solve for x

$$x=\frac{\sqrt{15142}}{98}+\frac{123}{49}\approx 3.765845247$$

$$x=-\frac{\sqrt{15142}}{98}+\frac{123}{49}\approx 1.254562917$$

Steps Using the Quadratic Formula

Multiply 0.5 and 9.8 to get 4.9.

$$23.15=24.6x-4.9x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24.6x-4.9x^{2}=23.15$$

Subtract 23.15 from both sides.

$$24.6x-4.9x^{2}-23.15=0$$

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.

$$-4.9x^{2}+24.6x-23.15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 24.6 for b, and -23.15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24.6±\sqrt{24.6^{2}-4\left(-4.9\right)\left(-23.15\right)}}{2\left(-4.9\right)}$$

Square 24.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-24.6±\sqrt{605.16-4\left(-4.9\right)\left(-23.15\right)}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$x=\frac{-24.6±\sqrt{605.16+19.6\left(-23.15\right)}}{2\left(-4.9\right)}$$

Multiply 19.6 times -23.15 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-24.6±\sqrt{605.16-453.74}}{2\left(-4.9\right)}$$

Add 605.16 to -453.74 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-24.6±\sqrt{151.42}}{2\left(-4.9\right)}$$

Take the square root of 151.42.

$$x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{-9.8}$$

Now solve the equation x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{-9.8} when ± is plus. Add -24.6 to \frac{\sqrt{15142}}{10}.

$$x=\frac{\frac{\sqrt{15142}}{10}-\frac{123}{5}}{-9.8}$$

Divide -\frac{123}{5}+\frac{\sqrt{15142}}{10} by -9.8 by multiplying -\frac{123}{5}+\frac{\sqrt{15142}}{10} by the reciprocal of -9.8.

$$x=-\frac{\sqrt{15142}}{98}+\frac{123}{49}$$

Now solve the equation x=\frac{-24.6±\frac{\sqrt{15142}}{10}}{-9.8} when ± is minus. Subtract \frac{\sqrt{15142}}{10} from -24.6.

$$x=\frac{-\frac{\sqrt{15142}}{10}-\frac{123}{5}}{-9.8}$$

Divide -\frac{123}{5}-\frac{\sqrt{15142}}{10} by -9.8 by multiplying -\frac{123}{5}-\frac{\sqrt{15142}}{10} by the reciprocal of -9.8.

$$x=\frac{\sqrt{15142}}{98}+\frac{123}{49}$$

The equation is now solved.

$$x=-\frac{\sqrt{15142}}{98}+\frac{123}{49} x=\frac{\sqrt{15142}}{98}+\frac{123}{49}$$

Steps for Completing the Square

Multiply 0.5 and 9.8 to get 4.9.

$$23.15=24.6x-4.9x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24.6x-4.9x^{2}=23.15$$

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.

$$-4.9x^{2}+24.6x=23.15$$

Divide both sides of the equation by -4.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-4.9x^{2}+24.6x}{-4.9}=\frac{23.15}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$x^{2}+\frac{24.6}{-4.9}x=\frac{23.15}{-4.9}$$

Divide 24.6 by -4.9 by multiplying 24.6 by the reciprocal of -4.9.

$$x^{2}-\frac{246}{49}x=\frac{23.15}{-4.9}$$

Divide 23.15 by -4.9 by multiplying 23.15 by the reciprocal of -4.9.

$$x^{2}-\frac{246}{49}x=-\frac{463}{98}$$

Divide -\frac{246}{49}, the coefficient of the x term, by 2 to get -\frac{123}{49}. Then add the square of -\frac{123}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{246}{49}x+\left(-\frac{123}{49}\right)^{2}=-\frac{463}{98}+\left(-\frac{123}{49}\right)^{2}$$

Square -\frac{123}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{246}{49}x+\frac{15129}{2401}=-\frac{463}{98}+\frac{15129}{2401}$$

Add -\frac{463}{98} to \frac{15129}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{246}{49}x+\frac{15129}{2401}=\frac{7571}{4802}$$

Factor x^{2}-\frac{246}{49}x+\frac{15129}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{123}{49}\right)^{2}=\frac{7571}{4802}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{123}{49}\right)^{2}}=\sqrt{\frac{7571}{4802}}$$

Simplify.

$$x-\frac{123}{49}=\frac{\sqrt{15142}}{98} x-\frac{123}{49}=-\frac{\sqrt{15142}}{98}$$

Add \frac{123}{49} to both sides of the equation.

$$x=\frac{\sqrt{15142}}{98}+\frac{123}{49} x=-\frac{\sqrt{15142}}{98}+\frac{123}{49}$$