Solve 6t+3.5t^2/2=12

$$6t+ \frac{ 3.5 { t }^{ 2 } }{ 2 } = 12$$

Solve for t

$$t = \frac{4 \sqrt{30} - 12}{7} \approx 1.415557471$$

$$t=\frac{-4\sqrt{30}-12}{7}\approx -4.8441289$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$12t+3.5t^{2}=24$$

Subtract 24 from both sides.

$$12t+3.5t^{2}-24=0$$

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.

$$3.5t^{2}+12t-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3.5 for a, 12 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-12±\sqrt{12^{2}-4\times 3.5\left(-24\right)}}{2\times 3.5}$$

Square 12.

$$t=\frac{-12±\sqrt{144-4\times 3.5\left(-24\right)}}{2\times 3.5}$$

Multiply -4 times 3.5.

$$t=\frac{-12±\sqrt{144-14\left(-24\right)}}{2\times 3.5}$$

Multiply -14 times -24.

$$t=\frac{-12±\sqrt{144+336}}{2\times 3.5}$$

Add 144 to 336.

$$t=\frac{-12±\sqrt{480}}{2\times 3.5}$$

Take the square root of 480.

$$t=\frac{-12±4\sqrt{30}}{2\times 3.5}$$

Multiply 2 times 3.5.

$$t=\frac{-12±4\sqrt{30}}{7}$$

Now solve the equation t=\frac{-12±4\sqrt{30}}{7} when ± is plus. Add -12 to 4\sqrt{30}.

$$t=\frac{4\sqrt{30}-12}{7}$$

Now solve the equation t=\frac{-12±4\sqrt{30}}{7} when ± is minus. Subtract 4\sqrt{30} from -12.

$$t=\frac{-4\sqrt{30}-12}{7}$$

The equation is now solved.

$$t=\frac{4\sqrt{30}-12}{7} t=\frac{-4\sqrt{30}-12}{7}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$12t+3.5t^{2}=24$$

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.

$$3.5t^{2}+12t=24$$

Divide both sides of the equation by 3.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3.5t^{2}+12t}{3.5}=\frac{24}{3.5}$$

Dividing by 3.5 undoes the multiplication by 3.5.

$$t^{2}+\frac{12}{3.5}t=\frac{24}{3.5}$$

Divide 12 by 3.5 by multiplying 12 by the reciprocal of 3.5.

$$t^{2}+\frac{24}{7}t=\frac{24}{3.5}$$

Divide 24 by 3.5 by multiplying 24 by the reciprocal of 3.5.

$$t^{2}+\frac{24}{7}t=\frac{48}{7}$$

Divide \frac{24}{7}, the coefficient of the x term, by 2 to get \frac{12}{7}. Then add the square of \frac{12}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{24}{7}t+\frac{12}{7}^{2}=\frac{48}{7}+\frac{12}{7}^{2}$$

Square \frac{12}{7} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{24}{7}t+\frac{144}{49}=\frac{48}{7}+\frac{144}{49}$$

Add \frac{48}{7} to \frac{144}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{24}{7}t+\frac{144}{49}=\frac{480}{49}$$

Factor t^{2}+\frac{24}{7}t+\frac{144}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{12}{7}\right)^{2}=\frac{480}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{12}{7}\right)^{2}}=\sqrt{\frac{480}{49}}$$

Simplify.

$$t+\frac{12}{7}=\frac{4\sqrt{30}}{7} t+\frac{12}{7}=-\frac{4\sqrt{30}}{7}$$

Subtract \frac{12}{7} from both sides of the equation.

$$t=\frac{4\sqrt{30}-12}{7} t=\frac{-4\sqrt{30}-12}{7}$$