Solve Using the Square Root Property 125=0.8x^2-2.4x

$125 = 0.8 x^{2} - 2.4 x$

Rewrite the equation as $0.8 x^{2} - 2.4 x = 125$ .

$0.8 x^{2} - 2.4 x = 125$

Move $125$ to the left side of the equation by subtracting it from both sides.

$0.8 x^{2} - 2.4 x - 125 = 0$

Factor $0.2$ out of $0.8 x^{2} - 2.4 x - 125$ .

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Factor $0.2$ out of $0.8 x^{2}$ .

$0.2 (4 x^{2}) - 2.4 x - 125 = 0$

Factor $0.2$ out of $- 2.4 x$ .

$0.2 (4 x^{2}) + 0.2 (- 12 x) - 125 = 0$

Factor $0.2$ out of $- 125$ .

$0.2 (4 x^{2}) + 0.2 (- 12 x) + 0.2 (- 625) = 0$

Factor $0.2$ out of $0.2 (4 x^{2}) + 0.2 (- 12 x)$ .

$0.2 (4 x^{2} - 12 x) + 0.2 (- 625) = 0$

Factor $0.2$ out of $0.2 (4 x^{2} - 12 x) + 0.2 (- 625)$ .

$0.2 (4 x^{2} - 12 x - 625) = 0$

Divide each term by $0.2$ and simplify.

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Divide each term in $0.2 (4 x^{2} - 12 x - 625) = 0$ by $0.2$ .

$\frac{0.2 (4 x^{2} - 12 x - 625)}{0.2} = \frac{0}{0.2}$

Cancel the common factor of $0.2$ .

$4 x^{2} - 12 x - 625 = \frac{0}{0.2}$

Divide $0$ by $0.2$ .

$4 x^{2} - 12 x - 625 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 4$ , $b = - 12$ , and $c = - 625$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (4 \cdot - 625)}}{2 \cdot 4}$

Simplify.

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Simplify the numerator.

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Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot (4 \cdot - 625)}}{2 \cdot 4}$

Multiply $4$ by $- 625$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 2500}}{2 \cdot 4}$

Multiply $- 4$ by $- 2500$ .

$x = \frac{12 \pm \sqrt{144 + 10000}}{2 \cdot 4}$

Add $144$ and $10000$ .

$x = \frac{12 \pm \sqrt{10144}}{2 \cdot 4}$

Rewrite $10144$ as $4^{2} \cdot 634$ .

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Factor $16$ out of $10144$ .

$x = \frac{12 \pm \sqrt{16 (634)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 634}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{634}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{12 \pm 4 \sqrt{634}}{8}$

Simplify $\frac{12 \pm 4 \sqrt{634}}{8}$ .

$x = \frac{3 \pm \sqrt{634}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot (4 \cdot - 625)}}{2 \cdot 4}$

Multiply $4$ by $- 625$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 2500}}{2 \cdot 4}$

Multiply $- 4$ by $- 2500$ .

$x = \frac{12 \pm \sqrt{144 + 10000}}{2 \cdot 4}$

Add $144$ and $10000$ .

$x = \frac{12 \pm \sqrt{10144}}{2 \cdot 4}$

Rewrite $10144$ as $4^{2} \cdot 634$ .

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Factor $16$ out of $10144$ .

$x = \frac{12 \pm \sqrt{16 (634)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 634}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{634}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{12 \pm 4 \sqrt{634}}{8}$

Simplify $\frac{12 \pm 4 \sqrt{634}}{8}$ .

$x = \frac{3 \pm \sqrt{634}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{634}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot (4 \cdot - 625)}}{2 \cdot 4}$

Multiply $4$ by $- 625$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot - 2500}}{2 \cdot 4}$

Multiply $- 4$ by $- 2500$ .

$x = \frac{12 \pm \sqrt{144 + 10000}}{2 \cdot 4}$

Add $144$ and $10000$ .

$x = \frac{12 \pm \sqrt{10144}}{2 \cdot 4}$

Rewrite $10144$ as $4^{2} \cdot 634$ .

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Factor $16$ out of $10144$ .

$x = \frac{12 \pm \sqrt{16 (634)}}{2 \cdot 4}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 634}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{634}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{12 \pm 4 \sqrt{634}}{8}$

Simplify $\frac{12 \pm 4 \sqrt{634}}{8}$ .

$x = \frac{3 \pm \sqrt{634}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{634}}{2}$

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{634}}{2}, \frac{3 - \sqrt{634}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{634}}{2}, \frac{3 - \sqrt{634}}{2}$

Decimal Form:

$x = 14.08967831 \dots, - 11.08967831 \dots$

Solve Using the Square Root Property 125=0.8x^2-2.4x

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