Solve 0.4x^2-6.8x+48=24

$$0.4 { x }^{ 2 } -6.8x+48=24$$

Solve for x

$$x=5$$

$$x=12$$

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.

$$0.4x^{2}-6.8x+48=24$$

Subtract 24 from both sides of the equation.

$$0.4x^{2}-6.8x+48-24=24-24$$

Subtracting 24 from itself leaves 0.

$$0.4x^{2}-6.8x+48-24=0$$

Subtract 24 from 48.

$$0.4x^{2}-6.8x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.4 for a, -6.8 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6.8\right)±\sqrt{\left(-6.8\right)^{2}-4\times 0.4\times 24}}{2\times 0.4}$$

Square -6.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-6.8\right)±\sqrt{46.24-4\times 0.4\times 24}}{2\times 0.4}$$

Multiply -4 times 0.4.

$$x=\frac{-\left(-6.8\right)±\sqrt{46.24-1.6\times 24}}{2\times 0.4}$$

Multiply -1.6 times 24.

$$x=\frac{-\left(-6.8\right)±\sqrt{46.24-38.4}}{2\times 0.4}$$

Add 46.24 to -38.4 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-6.8\right)±\sqrt{7.84}}{2\times 0.4}$$

Take the square root of 7.84.

$$x=\frac{-\left(-6.8\right)±\frac{14}{5}}{2\times 0.4}$$

The opposite of -6.8 is 6.8.

$$x=\frac{6.8±\frac{14}{5}}{2\times 0.4}$$

Multiply 2 times 0.4.

$$x=\frac{6.8±\frac{14}{5}}{0.8}$$

Now solve the equation x=\frac{6.8±\frac{14}{5}}{0.8} when ± is plus. Add 6.8 to \frac{14}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{48}{5}}{0.8}$$

Divide \frac{48}{5} by 0.8 by multiplying \frac{48}{5} by the reciprocal of 0.8.

$$x=12$$

Now solve the equation x=\frac{6.8±\frac{14}{5}}{0.8} when ± is minus. Subtract \frac{14}{5} from 6.8 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{4}{0.8}$$

Divide 4 by 0.8 by multiplying 4 by the reciprocal of 0.8.

$$x=5$$

The equation is now solved.

$$x=12 x=5$$

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.

$$0.4x^{2}-6.8x+48=24$$

Subtract 48 from both sides of the equation.

$$0.4x^{2}-6.8x+48-48=24-48$$

Subtracting 48 from itself leaves 0.

$$0.4x^{2}-6.8x=24-48$$

Subtract 48 from 24.

$$0.4x^{2}-6.8x=-24$$

Divide both sides of the equation by 0.4, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.4x^{2}-6.8x}{0.4}=-\frac{24}{0.4}$$

Dividing by 0.4 undoes the multiplication by 0.4.

$$x^{2}+\left(-\frac{6.8}{0.4}\right)x=-\frac{24}{0.4}$$

Divide -6.8 by 0.4 by multiplying -6.8 by the reciprocal of 0.4.

$$x^{2}-17x=-\frac{24}{0.4}$$

Divide -24 by 0.4 by multiplying -24 by the reciprocal of 0.4.

$$x^{2}-17x=-60$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-60+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-60+\frac{289}{4}$$

Add -60 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{49}{4}$$

Factor x^{2}-17x+\frac{289}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{7}{2} x-\frac{17}{2}=-\frac{7}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=12 x=5$$