Solve 32x^2-104x+73=0

$$32 { x }^{ 2 } -104x+73=0$$

Solve for x

$$x = \frac{\sqrt{23} + 13}{8} \approx 2.22447894$$

$$x = \frac{13 - \sqrt{23}}{8} \approx 1.02552106$$

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.

$$32x^{2}-104x+73=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 32 for a, -104 for b, and 73 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-104\right)±\sqrt{\left(-104\right)^{2}-4\times 32\times 73}}{2\times 32}$$

Square -104.

$$x=\frac{-\left(-104\right)±\sqrt{10816-4\times 32\times 73}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-104\right)±\sqrt{10816-128\times 73}}{2\times 32}$$

Multiply -128 times 73.

$$x=\frac{-\left(-104\right)±\sqrt{10816-9344}}{2\times 32}$$

Add 10816 to -9344.

$$x=\frac{-\left(-104\right)±\sqrt{1472}}{2\times 32}$$

Take the square root of 1472.

$$x=\frac{-\left(-104\right)±8\sqrt{23}}{2\times 32}$$

The opposite of -104 is 104.

$$x=\frac{104±8\sqrt{23}}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{104±8\sqrt{23}}{64}$$

Now solve the equation x=\frac{104±8\sqrt{23}}{64} when ± is plus. Add 104 to 8\sqrt{23}.

$$x=\frac{8\sqrt{23}+104}{64}$$

Divide 104+8\sqrt{23} by 64.

$$x=\frac{\sqrt{23}+13}{8}$$

Now solve the equation x=\frac{104±8\sqrt{23}}{64} when ± is minus. Subtract 8\sqrt{23} from 104.

$$x=\frac{104-8\sqrt{23}}{64}$$

Divide 104-8\sqrt{23} by 64.

$$x=\frac{13-\sqrt{23}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{23}+13}{8} x=\frac{13-\sqrt{23}}{8}$$

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.

$$32x^{2}-104x+73=0$$

Subtract 73 from both sides of the equation.

$$32x^{2}-104x+73-73=-73$$

Subtracting 73 from itself leaves 0.

$$32x^{2}-104x=-73$$

Divide both sides by 32.

$$\frac{32x^{2}-104x}{32}=-\frac{73}{32}$$

Dividing by 32 undoes the multiplication by 32.

$$x^{2}+\left(-\frac{104}{32}\right)x=-\frac{73}{32}$$

Reduce the fraction \frac{-104}{32} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{13}{4}x=-\frac{73}{32}$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=-\frac{73}{32}+\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=-\frac{73}{32}+\frac{169}{64}$$

Add -\frac{73}{32} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{23}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{23}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{23}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{\sqrt{23}}{8} x-\frac{13}{8}=-\frac{\sqrt{23}}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=\frac{\sqrt{23}+13}{8} x=\frac{13-\sqrt{23}}{8}$$