Solve 24x^2+10.8x=15.36

$$24 { x }^{ 2 } +10.8x=15.36$$

Solve for x

$$x=\frac{\sqrt{1105}-9}{40}\approx 0.606038507$$

$$x=\frac{-\sqrt{1105}-9}{40}\approx -1.056038507$$

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.

$$24x^{2}+10.8x=15.36$$

Subtract 15.36 from both sides of the equation.

$$24x^{2}+10.8x-15.36=15.36-15.36$$

Subtracting 15.36 from itself leaves 0.

$$24x^{2}+10.8x-15.36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 10.8 for b, and -15.36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10.8±\sqrt{10.8^{2}-4\times 24\left(-15.36\right)}}{2\times 24}$$

Square 10.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-10.8±\sqrt{116.64-4\times 24\left(-15.36\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-10.8±\sqrt{116.64-96\left(-15.36\right)}}{2\times 24}$$

Multiply -96 times -15.36.

$$x=\frac{-10.8±\sqrt{\frac{2916+36864}{25}}}{2\times 24}$$

Add 116.64 to 1474.56 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-10.8±\sqrt{1591.2}}{2\times 24}$$

Take the square root of 1591.2.

$$x=\frac{-10.8±\frac{6\sqrt{1105}}{5}}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-10.8±\frac{6\sqrt{1105}}{5}}{48}$$

Now solve the equation x=\frac{-10.8±\frac{6\sqrt{1105}}{5}}{48} when ± is plus. Add -10.8 to \frac{6\sqrt{1105}}{5}.

$$x=\frac{6\sqrt{1105}-54}{5\times 48}$$

Divide \frac{-54+6\sqrt{1105}}{5} by 48.

$$x=\frac{\sqrt{1105}-9}{40}$$

Now solve the equation x=\frac{-10.8±\frac{6\sqrt{1105}}{5}}{48} when ± is minus. Subtract \frac{6\sqrt{1105}}{5} from -10.8.

$$x=\frac{-6\sqrt{1105}-54}{5\times 48}$$

Divide \frac{-54-6\sqrt{1105}}{5} by 48.

$$x=\frac{-\sqrt{1105}-9}{40}$$

The equation is now solved.

$$x=\frac{\sqrt{1105}-9}{40} x=\frac{-\sqrt{1105}-9}{40}$$

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.

$$24x^{2}+10.8x=15.36$$

Divide both sides by 24.

$$\frac{24x^{2}+10.8x}{24}=\frac{15.36}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\frac{10.8}{24}x=\frac{15.36}{24}$$

Divide 10.8 by 24.

$$x^{2}+0.45x=\frac{15.36}{24}$$

Divide 15.36 by 24.

$$x^{2}+0.45x=0.64$$

Divide 0.45, the coefficient of the x term, by 2 to get 0.225. Then add the square of 0.225 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.45x+0.225^{2}=0.64+0.225^{2}$$

Square 0.225 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.45x+0.050625=0.64+0.050625$$

Add 0.64 to 0.050625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.45x+0.050625=0.690625$$

Factor x^{2}+0.45x+0.050625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.225\right)^{2}=0.690625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.225\right)^{2}}=\sqrt{0.690625}$$

Simplify.

$$x+0.225=\frac{\sqrt{1105}}{40} x+0.225=-\frac{\sqrt{1105}}{40}$$

Subtract 0.225 from both sides of the equation.

$$x=\frac{\sqrt{1105}-9}{40} x=\frac{-\sqrt{1105}-9}{40}$$