Solve -t^2+2t+24

$$- t ^ { 2 } + 2 t + 24$$

Factor

$$-\left(t-6\right)\left(t+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -t^{2}+at+bt+24. To find a and b, set up a system to be solved.

$$a+b=2 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 2.

$$a=6 b=-4$$

Rewrite -t^{2}+2t+24 as \left(-t^{2}+6t\right)+\left(-4t+24\right).

$$\left(-t^{2}+6t\right)+\left(-4t+24\right)$$

Factor out -t in the first and -4 in the second group.

$$-t\left(t-6\right)-4\left(t-6\right)$$

Factor out common term t-6 by using distributive property.

$$\left(t-6\right)\left(-t-4\right)$$

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.

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

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.

$$t=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square 2.

$$t=\frac{-2±\sqrt{4-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-2±\sqrt{4+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$t=\frac{-2±\sqrt{4+96}}{2\left(-1\right)}$$

Add 4 to 96.

$$t=\frac{-2±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$t=\frac{-2±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-2±10}{-2}$$

Now solve the equation t=\frac{-2±10}{-2} when ± is plus. Add -2 to 10.

$$t=\frac{8}{-2}$$

Divide 8 by -2.

$$t=-4$$

Now solve the equation t=\frac{-2±10}{-2} when ± is minus. Subtract 10 from -2.

$$t=-\frac{12}{-2}$$

Divide -12 by -2.

$$t=6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4 for x_{1} and 6 for x_{2}.

$$-t^{2}+2t+24=-\left(t-\left(-4\right)\right)\left(t-6\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-t^{2}+2t+24=-\left(t+4\right)\left(t-6\right)$$

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.

$$x ^ 2 -2x -24 = 0$$

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

$$r + s = 2 rs = -24$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(1 - u) (1 + u) = -24$$

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

$$1 - u^2 = -24$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -24-1 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =1 - 5 = -4 s = 1 + 5 = 6$$

Evaluate

$$-\left(t-6\right)\left(t+4\right)$$