Solve 1152(x)=-16x^2+224x+1152

$$1152(x)=-16 { x }^{ 2 } +224x+1152$$

Solve for x (complex solution)

$$x=\sqrt{913}-29\approx 1.215889859$$

$$x=-\left(\sqrt{913}+29\right)\approx -59.215889859$$

Steps Using the Quadratic Formula

Add 16x^{2} to both sides.

$$1152x+16x^{2}=224x+1152$$

Subtract 224x from both sides.

$$1152x+16x^{2}-224x=1152$$

Combine 1152x and -224x to get 928x.

$$928x+16x^{2}=1152$$

Subtract 1152 from both sides.

$$928x+16x^{2}-1152=0$$

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.

$$16x^{2}+928x-1152=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 928 for b, and -1152 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-928±\sqrt{928^{2}-4\times 16\left(-1152\right)}}{2\times 16}$$

Square 928.

$$x=\frac{-928±\sqrt{861184-4\times 16\left(-1152\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-928±\sqrt{861184-64\left(-1152\right)}}{2\times 16}$$

Multiply -64 times -1152.

$$x=\frac{-928±\sqrt{861184+73728}}{2\times 16}$$

Add 861184 to 73728.

$$x=\frac{-928±\sqrt{934912}}{2\times 16}$$

Take the square root of 934912.

$$x=\frac{-928±32\sqrt{913}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-928±32\sqrt{913}}{32}$$

Now solve the equation x=\frac{-928±32\sqrt{913}}{32} when ± is plus. Add -928 to 32\sqrt{913}.

$$x=\frac{32\sqrt{913}-928}{32}$$

Divide -928+32\sqrt{913} by 32.

$$x=\sqrt{913}-29$$

Now solve the equation x=\frac{-928±32\sqrt{913}}{32} when ± is minus. Subtract 32\sqrt{913} from -928.

$$x=\frac{-32\sqrt{913}-928}{32}$$

Divide -928-32\sqrt{913} by 32.

$$x=-\sqrt{913}-29$$

The equation is now solved.

$$x=\sqrt{913}-29 x=-\sqrt{913}-29$$

Steps for Completing the Square

Add 16x^{2} to both sides.

$$1152x+16x^{2}=224x+1152$$

Subtract 224x from both sides.

$$1152x+16x^{2}-224x=1152$$

Combine 1152x and -224x to get 928x.

$$928x+16x^{2}=1152$$

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.

$$16x^{2}+928x=1152$$

Divide both sides by 16.

$$\frac{16x^{2}+928x}{16}=\frac{1152}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{928}{16}x=\frac{1152}{16}$$

Divide 928 by 16.

$$x^{2}+58x=\frac{1152}{16}$$

Divide 1152 by 16.

$$x^{2}+58x=72$$

Divide 58, the coefficient of the x term, by 2 to get 29. Then add the square of 29 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+58x+29^{2}=72+29^{2}$$

Square 29.

$$x^{2}+58x+841=72+841$$

Add 72 to 841.

$$x^{2}+58x+841=913$$

Factor x^{2}+58x+841. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+29\right)^{2}=913$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+29\right)^{2}}=\sqrt{913}$$

Simplify.

$$x+29=\sqrt{913} x+29=-\sqrt{913}$$

Subtract 29 from both sides of the equation.

$$x=\sqrt{913}-29 x=-\sqrt{913}-29$$

Solve for x

$$x=\sqrt{913}-29\approx 1.215889859$$

$$x=-\sqrt{913}-29\approx -59.215889859$$

Steps Using the Quadratic Formula

Add 16x^{2} to both sides.

$$1152x+16x^{2}=224x+1152$$

Subtract 224x from both sides.

$$1152x+16x^{2}-224x=1152$$

Combine 1152x and -224x to get 928x.

$$928x+16x^{2}=1152$$

Subtract 1152 from both sides.

$$928x+16x^{2}-1152=0$$

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.

$$16x^{2}+928x-1152=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 928 for b, and -1152 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-928±\sqrt{928^{2}-4\times 16\left(-1152\right)}}{2\times 16}$$

Square 928.

$$x=\frac{-928±\sqrt{861184-4\times 16\left(-1152\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-928±\sqrt{861184-64\left(-1152\right)}}{2\times 16}$$

Multiply -64 times -1152.

$$x=\frac{-928±\sqrt{861184+73728}}{2\times 16}$$

Add 861184 to 73728.

$$x=\frac{-928±\sqrt{934912}}{2\times 16}$$

Take the square root of 934912.

$$x=\frac{-928±32\sqrt{913}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-928±32\sqrt{913}}{32}$$

Now solve the equation x=\frac{-928±32\sqrt{913}}{32} when ± is plus. Add -928 to 32\sqrt{913}.

$$x=\frac{32\sqrt{913}-928}{32}$$

Divide -928+32\sqrt{913} by 32.

$$x=\sqrt{913}-29$$

Now solve the equation x=\frac{-928±32\sqrt{913}}{32} when ± is minus. Subtract 32\sqrt{913} from -928.

$$x=\frac{-32\sqrt{913}-928}{32}$$

Divide -928-32\sqrt{913} by 32.

$$x=-\sqrt{913}-29$$

The equation is now solved.

$$x=\sqrt{913}-29 x=-\sqrt{913}-29$$

Steps for Completing the Square

Add 16x^{2} to both sides.

$$1152x+16x^{2}=224x+1152$$

Subtract 224x from both sides.

$$1152x+16x^{2}-224x=1152$$

Combine 1152x and -224x to get 928x.

$$928x+16x^{2}=1152$$

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.

$$16x^{2}+928x=1152$$

Divide both sides by 16.

$$\frac{16x^{2}+928x}{16}=\frac{1152}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{928}{16}x=\frac{1152}{16}$$

Divide 928 by 16.

$$x^{2}+58x=\frac{1152}{16}$$

Divide 1152 by 16.

$$x^{2}+58x=72$$

Divide 58, the coefficient of the x term, by 2 to get 29. Then add the square of 29 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+58x+29^{2}=72+29^{2}$$

Square 29.

$$x^{2}+58x+841=72+841$$

Add 72 to 841.

$$x^{2}+58x+841=913$$

Factor x^{2}+58x+841. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+29\right)^{2}=913$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+29\right)^{2}}=\sqrt{913}$$

Simplify.

$$x+29=\sqrt{913} x+29=-\sqrt{913}$$

Subtract 29 from both sides of the equation.

$$x=\sqrt{913}-29 x=-\sqrt{913}-29$$