Solve 2*7=2*4t+8*8t^2

$$2 \cdot 7 = 2 \cdot 4 t + 8 \cdot 8 t ^ { 2 }$$

Solve for t

$$t=\frac{\sqrt{57}-1}{16}\approx 0.409364652$$

$$t=\frac{-\sqrt{57}-1}{16}\approx -0.534364652$$

Steps Using the Quadratic Formula

Multiply 2 and 7 to get 14.

$$14=2\times 4t+8\times 8t^{2}$$

Do the multiplications.

$$14=8t+64t^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$8t+64t^{2}=14$$

Subtract 14 from both sides.

$$8t+64t^{2}-14=0$$

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.

$$64t^{2}+8t-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, 8 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-8±\sqrt{8^{2}-4\times 64\left(-14\right)}}{2\times 64}$$

Square 8.

$$t=\frac{-8±\sqrt{64-4\times 64\left(-14\right)}}{2\times 64}$$

Multiply -4 times 64.

$$t=\frac{-8±\sqrt{64-256\left(-14\right)}}{2\times 64}$$

Multiply -256 times -14.

$$t=\frac{-8±\sqrt{64+3584}}{2\times 64}$$

Add 64 to 3584.

$$t=\frac{-8±\sqrt{3648}}{2\times 64}$$

Take the square root of 3648.

$$t=\frac{-8±8\sqrt{57}}{2\times 64}$$

Multiply 2 times 64.

$$t=\frac{-8±8\sqrt{57}}{128}$$

Now solve the equation t=\frac{-8±8\sqrt{57}}{128} when ± is plus. Add -8 to 8\sqrt{57}.

$$t=\frac{8\sqrt{57}-8}{128}$$

Divide -8+8\sqrt{57} by 128.

$$t=\frac{\sqrt{57}-1}{16}$$

Now solve the equation t=\frac{-8±8\sqrt{57}}{128} when ± is minus. Subtract 8\sqrt{57} from -8.

$$t=\frac{-8\sqrt{57}-8}{128}$$

Divide -8-8\sqrt{57} by 128.

$$t=\frac{-\sqrt{57}-1}{16}$$

The equation is now solved.

$$t=\frac{\sqrt{57}-1}{16} t=\frac{-\sqrt{57}-1}{16}$$

Steps for Completing the Square

Multiply 2 and 7 to get 14.

$$14=2\times 4t+8\times 8t^{2}$$

Do the multiplications.

$$14=8t+64t^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$8t+64t^{2}=14$$

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.

$$64t^{2}+8t=14$$

Divide both sides by 64.

$$\frac{64t^{2}+8t}{64}=\frac{14}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$t^{2}+\frac{8}{64}t=\frac{14}{64}$$

Reduce the fraction \frac{8}{64} to lowest terms by extracting and canceling out 8.

$$t^{2}+\frac{1}{8}t=\frac{14}{64}$$

Reduce the fraction \frac{14}{64} to lowest terms by extracting and canceling out 2.

$$t^{2}+\frac{1}{8}t=\frac{7}{32}$$

Divide \frac{1}{8}, the coefficient of the x term, by 2 to get \frac{1}{16}. Then add the square of \frac{1}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{1}{8}t+\left(\frac{1}{16}\right)^{2}=\frac{7}{32}+\left(\frac{1}{16}\right)^{2}$$

Square \frac{1}{16} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{1}{8}t+\frac{1}{256}=\frac{7}{32}+\frac{1}{256}$$

Add \frac{7}{32} to \frac{1}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{1}{8}t+\frac{1}{256}=\frac{57}{256}$$

Factor t^{2}+\frac{1}{8}t+\frac{1}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{1}{16}\right)^{2}=\frac{57}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{1}{16}\right)^{2}}=\sqrt{\frac{57}{256}}$$

Simplify.

$$t+\frac{1}{16}=\frac{\sqrt{57}}{16} t+\frac{1}{16}=-\frac{\sqrt{57}}{16}$$

Subtract \frac{1}{16} from both sides of the equation.

$$t=\frac{\sqrt{57}-1}{16} t=\frac{-\sqrt{57}-1}{16}$$