Solve 80x^2+415x-351=0

$$80 { x }^{ 2 } +415x-351=0$$

Solve for x

$$x=\frac{\sqrt{284545}}{160}-\frac{83}{32}\approx 0.740172474$$

$$x=-\frac{\sqrt{284545}}{160}-\frac{83}{32}\approx -5.927672474$$

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.

$$80x^{2}+415x-351=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 80 for a, 415 for b, and -351 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-415±\sqrt{415^{2}-4\times 80\left(-351\right)}}{2\times 80}$$

Square 415.

$$x=\frac{-415±\sqrt{172225-4\times 80\left(-351\right)}}{2\times 80}$$

Multiply -4 times 80.

$$x=\frac{-415±\sqrt{172225-320\left(-351\right)}}{2\times 80}$$

Multiply -320 times -351.

$$x=\frac{-415±\sqrt{172225+112320}}{2\times 80}$$

Add 172225 to 112320.

$$x=\frac{-415±\sqrt{284545}}{2\times 80}$$

Multiply 2 times 80.

$$x=\frac{-415±\sqrt{284545}}{160}$$

Now solve the equation x=\frac{-415±\sqrt{284545}}{160} when ± is plus. Add -415 to \sqrt{284545}.

$$x=\frac{\sqrt{284545}-415}{160}$$

Divide -415+\sqrt{284545} by 160.

$$x=\frac{\sqrt{284545}}{160}-\frac{83}{32}$$

Now solve the equation x=\frac{-415±\sqrt{284545}}{160} when ± is minus. Subtract \sqrt{284545} from -415.

$$x=\frac{-\sqrt{284545}-415}{160}$$

Divide -415-\sqrt{284545} by 160.

$$x=-\frac{\sqrt{284545}}{160}-\frac{83}{32}$$

The equation is now solved.

$$x=\frac{\sqrt{284545}}{160}-\frac{83}{32} x=-\frac{\sqrt{284545}}{160}-\frac{83}{32}$$

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.

$$80x^{2}+415x-351=0$$

Add 351 to both sides of the equation.

$$80x^{2}+415x-351-\left(-351\right)=-\left(-351\right)$$

Subtracting -351 from itself leaves 0.

$$80x^{2}+415x=-\left(-351\right)$$

Subtract -351 from 0.

$$80x^{2}+415x=351$$

Divide both sides by 80.

$$\frac{80x^{2}+415x}{80}=\frac{351}{80}$$

Dividing by 80 undoes the multiplication by 80.

$$x^{2}+\frac{415}{80}x=\frac{351}{80}$$

Reduce the fraction \frac{415}{80} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{83}{16}x=\frac{351}{80}$$

Divide \frac{83}{16}, the coefficient of the x term, by 2 to get \frac{83}{32}. Then add the square of \frac{83}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{83}{16}x+\left(\frac{83}{32}\right)^{2}=\frac{351}{80}+\left(\frac{83}{32}\right)^{2}$$

Square \frac{83}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{83}{16}x+\frac{6889}{1024}=\frac{351}{80}+\frac{6889}{1024}$$

Add \frac{351}{80} to \frac{6889}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{83}{16}x+\frac{6889}{1024}=\frac{56909}{5120}$$

Factor x^{2}+\frac{83}{16}x+\frac{6889}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{83}{32}\right)^{2}=\frac{56909}{5120}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{83}{32}\right)^{2}}=\sqrt{\frac{56909}{5120}}$$

Simplify.

$$x+\frac{83}{32}=\frac{\sqrt{284545}}{160} x+\frac{83}{32}=-\frac{\sqrt{284545}}{160}$$

Subtract \frac{83}{32} from both sides of the equation.

$$x=\frac{\sqrt{284545}}{160}-\frac{83}{32} x=-\frac{\sqrt{284545}}{160}-\frac{83}{32}$$