Solve -t^2+24t+1

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

Factor

$$-\left(t-\left(12-\sqrt{145}\right)\right)\left(t-\left(\sqrt{145}+12\right)\right)$$

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.

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

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.

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

Square 24.

$$t=\frac{-24±\sqrt{576-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-24±\sqrt{576+4}}{2\left(-1\right)}$$

Add 576 to 4.

$$t=\frac{-24±\sqrt{580}}{2\left(-1\right)}$$

Take the square root of 580.

$$t=\frac{-24±2\sqrt{145}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-24±2\sqrt{145}}{-2}$$

Now solve the equation t=\frac{-24±2\sqrt{145}}{-2} when ± is plus. Add -24 to 2\sqrt{145}.

$$t=\frac{2\sqrt{145}-24}{-2}$$

Divide -24+2\sqrt{145} by -2.

$$t=12-\sqrt{145}$$

Now solve the equation t=\frac{-24±2\sqrt{145}}{-2} when ± is minus. Subtract 2\sqrt{145} from -24.

$$t=\frac{-2\sqrt{145}-24}{-2}$$

Divide -24-2\sqrt{145} by -2.

$$t=\sqrt{145}+12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12-\sqrt{145} for x_{1} and 12+\sqrt{145} for x_{2}.

$$-t^{2}+24t+1=-\left(t-\left(12-\sqrt{145}\right)\right)\left(t-\left(\sqrt{145}+12\right)\right)$$

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.

$$x ^ 2 -24x -1 = 0$$

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

$$r + s = 24 rs = -1$$

Two numbers r and s sum up to 24 exactly when the average of the two numbers is \frac{1}{2}*24 = 12. 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>

$$r = 12 - u s = 12 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(12 - u) (12 + u) = -1$$

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

$$144 - u^2 = -1$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = -1-144 = -145$$

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

$$u^2 = 145 u = \pm\sqrt{145} = \pm \sqrt{145}$$

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

$$r =12 - \sqrt{145} = -0.042 s = 12 + \sqrt{145} = 24.042$$

Evaluate

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