Solve 43t^2+6t-15

$$43 t ^ { 2 } + 6 t - 15$$

Factor

$$43\left(t-\frac{-\sqrt{654}-3}{43}\right)\left(t-\frac{\sqrt{654}-3}{43}\right)$$

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.

$$43t^{2}+6t-15=0$$

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.

$$t=\frac{-6±\sqrt{6^{2}-4\times 43\left(-15\right)}}{2\times 43}$$

Square 6.

$$t=\frac{-6±\sqrt{36-4\times 43\left(-15\right)}}{2\times 43}$$

Multiply -4 times 43.

$$t=\frac{-6±\sqrt{36-172\left(-15\right)}}{2\times 43}$$

Multiply -172 times -15.

$$t=\frac{-6±\sqrt{36+2580}}{2\times 43}$$

Add 36 to 2580.

$$t=\frac{-6±\sqrt{2616}}{2\times 43}$$

Take the square root of 2616.

$$t=\frac{-6±2\sqrt{654}}{2\times 43}$$

Multiply 2 times 43.

$$t=\frac{-6±2\sqrt{654}}{86}$$

Now solve the equation t=\frac{-6±2\sqrt{654}}{86} when ± is plus. Add -6 to 2\sqrt{654}.

$$t=\frac{2\sqrt{654}-6}{86}$$

Divide -6+2\sqrt{654} by 86.

$$t=\frac{\sqrt{654}-3}{43}$$

Now solve the equation t=\frac{-6±2\sqrt{654}}{86} when ± is minus. Subtract 2\sqrt{654} from -6.

$$t=\frac{-2\sqrt{654}-6}{86}$$

Divide -6-2\sqrt{654} by 86.

$$t=\frac{-\sqrt{654}-3}{43}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{654}}{43} for x_{1} and \frac{-3-\sqrt{654}}{43} for x_{2}.

$$43t^{2}+6t-15=43\left(t-\frac{\sqrt{654}-3}{43}\right)\left(t-\frac{-\sqrt{654}-3}{43}\right)$$

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 43

$$x ^ 2 +\frac{6}{43}x -\frac{15}{43} = 0$$

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

$$r + s = -\frac{6}{43} rs = -\frac{15}{43}$$

Two numbers r and s sum up to -\frac{6}{43} exactly when the average of the two numbers is \frac{1}{2}*-\frac{6}{43} = -\frac{3}{43}. 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>

$$r = -\frac{3}{43} - u s = -\frac{3}{43} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{15}{43}

$$(-\frac{3}{43} - u) (-\frac{3}{43} + u) = -\frac{15}{43}$$

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

$$\frac{9}{1849} - u^2 = -\frac{15}{43}$$

Simplify the expression by subtracting \frac{9}{1849} on both sides

$$-u^2 = -\frac{15}{43}-\frac{9}{1849} = -\frac{654}{1849}$$

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

$$u^2 = \frac{654}{1849} u = \pm\sqrt{\frac{654}{1849}} = \pm \frac{\sqrt{654}}{43}$$

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

$$r =-\frac{3}{43} - \frac{\sqrt{654}}{43} = -0.664 s = -\frac{3}{43} + \frac{\sqrt{654}}{43} = 0.525$$

Evaluate

$$43t^{2}+6t-15$$