Solve 6t+35t^2/2=12

$$6t+ \frac{ 35 { t }^{ 2 } }{ 2 } = 12$$

Solve for t

$$t=\frac{2\sqrt{219}-6}{35}\approx 0.674208491$$

$$t=\frac{-2\sqrt{219}-6}{35}\approx -1.017065634$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$12t+35t^{2}=24$$

Subtract 24 from both sides.

$$12t+35t^{2}-24=0$$

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.

$$35t^{2}+12t-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, 12 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-12±\sqrt{12^{2}-4\times 35\left(-24\right)}}{2\times 35}$$

Square 12.

$$t=\frac{-12±\sqrt{144-4\times 35\left(-24\right)}}{2\times 35}$$

Multiply -4 times 35.

$$t=\frac{-12±\sqrt{144-140\left(-24\right)}}{2\times 35}$$

Multiply -140 times -24.

$$t=\frac{-12±\sqrt{144+3360}}{2\times 35}$$

Add 144 to 3360.

$$t=\frac{-12±\sqrt{3504}}{2\times 35}$$

Take the square root of 3504.

$$t=\frac{-12±4\sqrt{219}}{2\times 35}$$

Multiply 2 times 35.

$$t=\frac{-12±4\sqrt{219}}{70}$$

Now solve the equation t=\frac{-12±4\sqrt{219}}{70} when ± is plus. Add -12 to 4\sqrt{219}.

$$t=\frac{4\sqrt{219}-12}{70}$$

Divide -12+4\sqrt{219} by 70.

$$t=\frac{2\sqrt{219}-6}{35}$$

Now solve the equation t=\frac{-12±4\sqrt{219}}{70} when ± is minus. Subtract 4\sqrt{219} from -12.

$$t=\frac{-4\sqrt{219}-12}{70}$$

Divide -12-4\sqrt{219} by 70.

$$t=\frac{-2\sqrt{219}-6}{35}$$

The equation is now solved.

$$t=\frac{2\sqrt{219}-6}{35} t=\frac{-2\sqrt{219}-6}{35}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$12t+35t^{2}=24$$

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.

$$35t^{2}+12t=24$$

Divide both sides by 35.

$$\frac{35t^{2}+12t}{35}=\frac{24}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$t^{2}+\frac{12}{35}t=\frac{24}{35}$$

Divide \frac{12}{35}, the coefficient of the x term, by 2 to get \frac{6}{35}. Then add the square of \frac{6}{35} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{12}{35}t+\left(\frac{6}{35}\right)^{2}=\frac{24}{35}+\left(\frac{6}{35}\right)^{2}$$

Square \frac{6}{35} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{12}{35}t+\frac{36}{1225}=\frac{24}{35}+\frac{36}{1225}$$

Add \frac{24}{35} to \frac{36}{1225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{12}{35}t+\frac{36}{1225}=\frac{876}{1225}$$

Factor t^{2}+\frac{12}{35}t+\frac{36}{1225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{6}{35}\right)^{2}=\frac{876}{1225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{6}{35}\right)^{2}}=\sqrt{\frac{876}{1225}}$$

Simplify.

$$t+\frac{6}{35}=\frac{2\sqrt{219}}{35} t+\frac{6}{35}=-\frac{2\sqrt{219}}{35}$$

Subtract \frac{6}{35} from both sides of the equation.

$$t=\frac{2\sqrt{219}-6}{35} t=\frac{-2\sqrt{219}-6}{35}$$