Solve -k^2+7k-5

$$- k ^ { 2 } + 7 k - 5$$

Factor

$$-\left(k-\frac{7-\sqrt{29}}{2}\right)\left(k-\frac{\sqrt{29}+7}{2}\right)$$

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.

$$-k^{2}+7k-5=0$$

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.

$$k=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Square 7.

$$k=\frac{-7±\sqrt{49-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$k=\frac{-7±\sqrt{49+4\left(-5\right)}}{2\left(-1\right)}$$

Multiply 4 times -5.

$$k=\frac{-7±\sqrt{49-20}}{2\left(-1\right)}$$

Add 49 to -20.

$$k=\frac{-7±\sqrt{29}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$k=\frac{-7±\sqrt{29}}{-2}$$

Now solve the equation k=\frac{-7±\sqrt{29}}{-2} when ± is plus. Add -7 to \sqrt{29}.

$$k=\frac{\sqrt{29}-7}{-2}$$

Divide -7+\sqrt{29} by -2.

$$k=\frac{7-\sqrt{29}}{2}$$

Now solve the equation k=\frac{-7±\sqrt{29}}{-2} when ± is minus. Subtract \sqrt{29} from -7.

$$k=\frac{-\sqrt{29}-7}{-2}$$

Divide -7-\sqrt{29} by -2.

$$k=\frac{\sqrt{29}+7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{29}}{2} for x_{1} and \frac{7+\sqrt{29}}{2} for x_{2}.

$$-k^{2}+7k-5=-\left(k-\frac{7-\sqrt{29}}{2}\right)\left(k-\frac{\sqrt{29}+7}{2}\right)$$

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.

$$x ^ 2 -7x +5 = 0$$

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

$$r + s = 7 rs = 5$$

Two numbers r and s sum up to 7 exactly when the average of the two numbers is \frac{1}{2}*7 = \frac{7}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{7}{2} - u s = \frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 5

$$(\frac{7}{2} - u) (\frac{7}{2} + u) = 5$$

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

$$\frac{49}{4} - u^2 = 5$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 5-\frac{49}{4} = -\frac{29}{4}$$

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

$$u^2 = \frac{29}{4} u = \pm\sqrt{\frac{29}{4}} = \pm \frac{\sqrt{29}}{2}$$

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

$$r =\frac{7}{2} - \frac{\sqrt{29}}{2} = 0.807 s = \frac{7}{2} + \frac{\sqrt{29}}{2} = 6.193$$

Evaluate

$$-k^{2}+7k-5$$