Solve 4t^2-12t-5

$$4 t ^ { 2 } - 12 t - 5$$

Factor

$$4\left(t-\frac{3-\sqrt{14}}{2}\right)\left(t-\frac{\sqrt{14}+3}{2}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$4t^{2}-12t-5=0$$

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.

$$t=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-5\right)}}{2\times 4}$$

Square -12.

$$t=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-5\right)}}{2\times 4}$$

Multiply -4 times 4.

$$t=\frac{-\left(-12\right)±\sqrt{144-16\left(-5\right)}}{2\times 4}$$

Multiply -16 times -5.

$$t=\frac{-\left(-12\right)±\sqrt{144+80}}{2\times 4}$$

Add 144 to 80.

$$t=\frac{-\left(-12\right)±\sqrt{224}}{2\times 4}$$

Take the square root of 224.

$$t=\frac{-\left(-12\right)±4\sqrt{14}}{2\times 4}$$

The opposite of -12 is 12.

$$t=\frac{12±4\sqrt{14}}{2\times 4}$$

Multiply 2 times 4.

$$t=\frac{12±4\sqrt{14}}{8}$$

Now solve the equation t=\frac{12±4\sqrt{14}}{8} when ± is plus. Add 12 to 4\sqrt{14}.

$$t=\frac{4\sqrt{14}+12}{8}$$

Divide 12+4\sqrt{14} by 8.

$$t=\frac{\sqrt{14}+3}{2}$$

Now solve the equation t=\frac{12±4\sqrt{14}}{8} when ± is minus. Subtract 4\sqrt{14} from 12.

$$t=\frac{12-4\sqrt{14}}{8}$$

Divide 12-4\sqrt{14} by 8.

$$t=\frac{3-\sqrt{14}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{14}}{2} for x_{1} and \frac{3-\sqrt{14}}{2} for x_{2}.

$$4t^{2}-12t-5=4\left(t-\frac{\sqrt{14}+3}{2}\right)\left(t-\frac{3-\sqrt{14}}{2}\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 -3x -\frac{5}{4} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 3 rs = -\frac{5}{4}$$

Two numbers r and s sum up to 3 exactly when the average of the two numbers is \frac{1}{2}*3 = \frac{3}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{3}{2} - u s = \frac{3}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{4}

$$(\frac{3}{2} - u) (\frac{3}{2} + u) = -\frac{5}{4}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{9}{4} - u^2 = -\frac{5}{4}$$

Simplify the expression by subtracting \frac{9}{4} on both sides

$$-u^2 = -\frac{5}{4}-\frac{9}{4} = -\frac{7}{2}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{7}{2} u = \pm\sqrt{\frac{7}{2}} = \pm \frac{\sqrt{7}}{\sqrt{2}}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{3}{2} - \frac{\sqrt{7}}{\sqrt{2}} = -0.371 s = \frac{3}{2} + \frac{\sqrt{7}}{\sqrt{2}} = 3.371$$

Evaluate

$$4t^{2}-12t-5$$