Solve 4x+52x^2=85-25x

$$4x+52 { x }^{ 2 } = 85-25x$$

Solve for x

$$x = \frac{\sqrt{18521} - 29}{104} \approx 1.029729622$$

$$x=\frac{-\sqrt{18521}-29}{104}\approx -1.58742193$$

Steps Using the Quadratic Formula

Subtract 85 from both sides.

$$4x+52x^{2}-85=-25x$$

Add 25x to both sides.

$$4x+52x^{2}-85+25x=0$$

Combine 4x and 25x to get 29x.

$$29x+52x^{2}-85=0$$

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.

$$52x^{2}+29x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 52 for a, 29 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-29±\sqrt{29^{2}-4\times 52\left(-85\right)}}{2\times 52}$$

Square 29.

$$x=\frac{-29±\sqrt{841-4\times 52\left(-85\right)}}{2\times 52}$$

Multiply -4 times 52.

$$x=\frac{-29±\sqrt{841-208\left(-85\right)}}{2\times 52}$$

Multiply -208 times -85.

$$x=\frac{-29±\sqrt{841+17680}}{2\times 52}$$

Add 841 to 17680.

$$x=\frac{-29±\sqrt{18521}}{2\times 52}$$

Multiply 2 times 52.

$$x=\frac{-29±\sqrt{18521}}{104}$$

Now solve the equation x=\frac{-29±\sqrt{18521}}{104} when ± is plus. Add -29 to \sqrt{18521}.

$$x=\frac{\sqrt{18521}-29}{104}$$

Now solve the equation x=\frac{-29±\sqrt{18521}}{104} when ± is minus. Subtract \sqrt{18521} from -29.

$$x=\frac{-\sqrt{18521}-29}{104}$$

The equation is now solved.

$$x=\frac{\sqrt{18521}-29}{104} x=\frac{-\sqrt{18521}-29}{104}$$

Steps for Completing the Square

Add 25x to both sides.

$$4x+52x^{2}+25x=85$$

Combine 4x and 25x to get 29x.

$$29x+52x^{2}=85$$

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.

$$52x^{2}+29x=85$$

Divide both sides by 52.

$$\frac{52x^{2}+29x}{52}=\frac{85}{52}$$

Dividing by 52 undoes the multiplication by 52.

$$x^{2}+\frac{29}{52}x=\frac{85}{52}$$

Divide \frac{29}{52}, the coefficient of the x term, by 2 to get \frac{29}{104}. Then add the square of \frac{29}{104} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{29}{52}x+\left(\frac{29}{104}\right)^{2}=\frac{85}{52}+\left(\frac{29}{104}\right)^{2}$$

Square \frac{29}{104} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{29}{52}x+\frac{841}{10816}=\frac{85}{52}+\frac{841}{10816}$$

Add \frac{85}{52} to \frac{841}{10816} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{29}{52}x+\frac{841}{10816}=\frac{18521}{10816}$$

Factor x^{2}+\frac{29}{52}x+\frac{841}{10816}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{29}{104}\right)^{2}=\frac{18521}{10816}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{29}{104}\right)^{2}}=\sqrt{\frac{18521}{10816}}$$

Simplify.

$$x+\frac{29}{104}=\frac{\sqrt{18521}}{104} x+\frac{29}{104}=-\frac{\sqrt{18521}}{104}$$

Subtract \frac{29}{104} from both sides of the equation.

$$x=\frac{\sqrt{18521}-29}{104} x=\frac{-\sqrt{18521}-29}{104}$$