Solve 10-t^2+7=4t

$$10 - t ^ { 2 } + 7 = 4 t$$

Solve for t (complex solution)

$$t=\sqrt{21}-2\approx 2.582575695$$

$$t=-\left(\sqrt{21}+2\right)\approx -6.582575695$$

Steps Using the Quadratic Formula

Add 10 and 7 to get 17.

$$17-t^{2}=4t$$

Subtract 4t from both sides.

$$17-t^{2}-4t=0$$

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.

$$-t^{2}-4t+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 17}}{2\left(-1\right)}$$

Square -4.

$$t=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 17}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-\left(-4\right)±\sqrt{16+4\times 17}}{2\left(-1\right)}$$

Multiply 4 times 17.

$$t=\frac{-\left(-4\right)±\sqrt{16+68}}{2\left(-1\right)}$$

Add 16 to 68.

$$t=\frac{-\left(-4\right)±\sqrt{84}}{2\left(-1\right)}$$

Take the square root of 84.

$$t=\frac{-\left(-4\right)±2\sqrt{21}}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$t=\frac{4±2\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{4±2\sqrt{21}}{-2}$$

Now solve the equation t=\frac{4±2\sqrt{21}}{-2} when ± is plus. Add 4 to 2\sqrt{21}.

$$t=\frac{2\sqrt{21}+4}{-2}$$

Divide 4+2\sqrt{21} by -2.

$$t=-\left(\sqrt{21}+2\right)$$

Now solve the equation t=\frac{4±2\sqrt{21}}{-2} when ± is minus. Subtract 2\sqrt{21} from 4.

$$t=\frac{4-2\sqrt{21}}{-2}$$

Divide 4-2\sqrt{21} by -2.

$$t=\sqrt{21}-2$$

The equation is now solved.

$$t=-\left(\sqrt{21}+2\right) t=\sqrt{21}-2$$

Steps for Completing the Square

Add 10 and 7 to get 17.

$$17-t^{2}=4t$$

Subtract 4t from both sides.

$$17-t^{2}-4t=0$$

Subtract 17 from both sides. Anything subtracted from zero gives its negation.

$$-t^{2}-4t=-17$$

Divide both sides by -1.

$$\frac{-t^{2}-4t}{-1}=-\frac{17}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\left(-\frac{4}{-1}\right)t=-\frac{17}{-1}$$

Divide -4 by -1.

$$t^{2}+4t=-\frac{17}{-1}$$

Divide -17 by -1.

$$t^{2}+4t=17$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+4t+2^{2}=17+2^{2}$$

Square 2.

$$t^{2}+4t+4=17+4$$

Add 17 to 4.

$$t^{2}+4t+4=21$$

Factor t^{2}+4t+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+2\right)^{2}=21$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+2\right)^{2}}=\sqrt{21}$$

Simplify.

$$t+2=\sqrt{21} t+2=-\sqrt{21}$$

Subtract 2 from both sides of the equation.

$$t=\sqrt{21}-2 t=-\sqrt{21}-2$$

Solve for t

$$t=\sqrt{21}-2\approx 2.582575695$$

$$t=-\sqrt{21}-2\approx -6.582575695$$

Steps Using the Quadratic Formula

Add 10 and 7 to get 17.

$$17-t^{2}=4t$$

Subtract 4t from both sides.

$$17-t^{2}-4t=0$$

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.

$$-t^{2}-4t+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 17}}{2\left(-1\right)}$$

Square -4.

$$t=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 17}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-\left(-4\right)±\sqrt{16+4\times 17}}{2\left(-1\right)}$$

Multiply 4 times 17.

$$t=\frac{-\left(-4\right)±\sqrt{16+68}}{2\left(-1\right)}$$

Add 16 to 68.

$$t=\frac{-\left(-4\right)±\sqrt{84}}{2\left(-1\right)}$$

Take the square root of 84.

$$t=\frac{-\left(-4\right)±2\sqrt{21}}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$t=\frac{4±2\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{4±2\sqrt{21}}{-2}$$

Now solve the equation t=\frac{4±2\sqrt{21}}{-2} when ± is plus. Add 4 to 2\sqrt{21}.

$$t=\frac{2\sqrt{21}+4}{-2}$$

Divide 4+2\sqrt{21} by -2.

$$t=-\left(\sqrt{21}+2\right)$$

Now solve the equation t=\frac{4±2\sqrt{21}}{-2} when ± is minus. Subtract 2\sqrt{21} from 4.

$$t=\frac{4-2\sqrt{21}}{-2}$$

Divide 4-2\sqrt{21} by -2.

$$t=\sqrt{21}-2$$

The equation is now solved.

$$t=-\left(\sqrt{21}+2\right) t=\sqrt{21}-2$$

Steps for Completing the Square

Add 10 and 7 to get 17.

$$17-t^{2}=4t$$

Subtract 4t from both sides.

$$17-t^{2}-4t=0$$

Subtract 17 from both sides. Anything subtracted from zero gives its negation.

$$-t^{2}-4t=-17$$

Divide both sides by -1.

$$\frac{-t^{2}-4t}{-1}=-\frac{17}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\left(-\frac{4}{-1}\right)t=-\frac{17}{-1}$$

Divide -4 by -1.

$$t^{2}+4t=-\frac{17}{-1}$$

Divide -17 by -1.

$$t^{2}+4t=17$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+4t+2^{2}=17+2^{2}$$

Square 2.

$$t^{2}+4t+4=17+4$$

Add 17 to 4.

$$t^{2}+4t+4=21$$

Factor t^{2}+4t+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+2\right)^{2}=21$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+2\right)^{2}}=\sqrt{21}$$

Simplify.

$$t+2=\sqrt{21} t+2=-\sqrt{21}$$

Subtract 2 from both sides of the equation.

$$t=\sqrt{21}-2 t=-\sqrt{21}-2$$