Solve 3t^2+24t+1

$$3 t ^ { 2 } + 24 t + 1$$

Factor

$$3\left(t-\left(-\frac{\sqrt{141}}{3}-4\right)\right)\left(t-\left(\frac{\sqrt{141}}{3}-4\right)\right)$$

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.

$$3t^{2}+24t+1=0$$

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.

$$t=\frac{-24±\sqrt{24^{2}-4\times 3}}{2\times 3}$$

Square 24.

$$t=\frac{-24±\sqrt{576-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$t=\frac{-24±\sqrt{576-12}}{2\times 3}$$

Add 576 to -12.

$$t=\frac{-24±\sqrt{564}}{2\times 3}$$

Take the square root of 564.

$$t=\frac{-24±2\sqrt{141}}{2\times 3}$$

Multiply 2 times 3.

$$t=\frac{-24±2\sqrt{141}}{6}$$

Now solve the equation t=\frac{-24±2\sqrt{141}}{6} when ± is plus. Add -24 to 2\sqrt{141}.

$$t=\frac{2\sqrt{141}-24}{6}$$

Divide -24+2\sqrt{141} by 6.

$$t=\frac{\sqrt{141}}{3}-4$$

Now solve the equation t=\frac{-24±2\sqrt{141}}{6} when ± is minus. Subtract 2\sqrt{141} from -24.

$$t=\frac{-2\sqrt{141}-24}{6}$$

Divide -24-2\sqrt{141} by 6.

$$t=-\frac{\sqrt{141}}{3}-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4+\frac{\sqrt{141}}{3} for x_{1} and -4-\frac{\sqrt{141}}{3} for x_{2}.

$$3t^{2}+24t+1=3\left(t-\left(\frac{\sqrt{141}}{3}-4\right)\right)\left(t-\left(-\frac{\sqrt{141}}{3}-4\right)\right)$$

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 3

$$x ^ 2 +8x +\frac{1}{3} = 0$$

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

$$r + s = -8 rs = \frac{1}{3}$$

Two numbers r and s sum up to -8 exactly when the average of the two numbers is \frac{1}{2}*-8 = -4. 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>

$$r = -4 - u s = -4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{3}

$$(-4 - u) (-4 + u) = \frac{1}{3}$$

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

$$16 - u^2 = \frac{1}{3}$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = \frac{1}{3}-16 = -\frac{47}{3}$$

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

$$u^2 = \frac{47}{3} u = \pm\sqrt{\frac{47}{3}} = \pm \frac{\sqrt{47}}{\sqrt{3}}$$

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

$$r =-4 - \frac{\sqrt{47}}{\sqrt{3}} = -7.958 s = -4 + \frac{\sqrt{47}}{\sqrt{3}} = -0.042$$

Evaluate

$$3t^{2}+24t+1$$