Solve 0=7x^2-100x+288

$$0 = 7 x ^ { 2 } - 100 x + 288$$

Solve for x

$$x=4$$

$$x = \frac{72}{7} = 10\frac{2}{7} \approx 10.285714286$$

Steps Using the Quadratic Formula

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

$$7x^{2}-100x+288=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -100 for b, and 288 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 7\times 288}}{2\times 7}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\times 7\times 288}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-100\right)±\sqrt{10000-28\times 288}}{2\times 7}$$

Multiply -28 times 288.

$$x=\frac{-\left(-100\right)±\sqrt{10000-8064}}{2\times 7}$$

Add 10000 to -8064.

$$x=\frac{-\left(-100\right)±\sqrt{1936}}{2\times 7}$$

Take the square root of 1936.

$$x=\frac{-\left(-100\right)±44}{2\times 7}$$

The opposite of -100 is 100.

$$x=\frac{100±44}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{100±44}{14}$$

Now solve the equation x=\frac{100±44}{14} when ± is plus. Add 100 to 44.

$$x=\frac{144}{14}$$

Reduce the fraction \frac{144}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{72}{7}$$

Now solve the equation x=\frac{100±44}{14} when ± is minus. Subtract 44 from 100.

$$x=\frac{56}{14}$$

Divide 56 by 14.

$$x=4$$

The equation is now solved.

$$x=\frac{72}{7} x=4$$

Steps for Completing the Square

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

$$7x^{2}-100x+288=0$$

Subtract 288 from both sides. Anything subtracted from zero gives its negation.

$$7x^{2}-100x=-288$$

Divide both sides by 7.

$$\frac{7x^{2}-100x}{7}=-\frac{288}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{100}{7}x=-\frac{288}{7}$$

Divide -\frac{100}{7}, the coefficient of the x term, by 2 to get -\frac{50}{7}. Then add the square of -\frac{50}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{100}{7}x+\left(-\frac{50}{7}\right)^{2}=-\frac{288}{7}+\left(-\frac{50}{7}\right)^{2}$$

Square -\frac{50}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{100}{7}x+\frac{2500}{49}=-\frac{288}{7}+\frac{2500}{49}$$

Add -\frac{288}{7} to \frac{2500}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{100}{7}x+\frac{2500}{49}=\frac{484}{49}$$

Factor x^{2}-\frac{100}{7}x+\frac{2500}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{50}{7}\right)^{2}=\frac{484}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{50}{7}\right)^{2}}=\sqrt{\frac{484}{49}}$$

Simplify.

$$x-\frac{50}{7}=\frac{22}{7} x-\frac{50}{7}=-\frac{22}{7}$$

Add \frac{50}{7} to both sides of the equation.

$$x=\frac{72}{7} x=4$$