Basic Math Examples

$4 d^{2} - 5 d = 21 + y$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $21$ from both sides of the equation.

$4 d^{2} - 5 d - 21 = y$

Step 1.2

Subtract $y$ from both sides of the equation.

$4 d^{2} - 5 d - 21 - y = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 4$ , $b = - 5$ , and $c = - 21 - y$ into the quadratic formula and solve for $d$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (4 \cdot (- 21 - y))}}{2 \cdot 4}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 5$ to the power of $2$ .

$d = \frac{5 \pm \sqrt{25 - 4 \cdot 4 \cdot (- 21 - y)}}{2 \cdot 4}$

Step 4.1.2

Multiply $- 4$ by $4$ .

$d = \frac{5 \pm \sqrt{25 - 16 \cdot (- 21 - y)}}{2 \cdot 4}$

Step 4.1.3

Apply the distributive property.

$d = \frac{5 \pm \sqrt{25 - 16 \cdot - 21 - 16 (- y)}}{2 \cdot 4}$

Step 4.1.4

Multiply $- 16$ by $- 21$ .

$d = \frac{5 \pm \sqrt{25 + 336 - 16 (- y)}}{2 \cdot 4}$

Step 4.1.5

Multiply $- 1$ by $- 16$ .

$d = \frac{5 \pm \sqrt{25 + 336 + 16 y}}{2 \cdot 4}$

Step 4.1.6

Add $25$ and $336$ .

$d = \frac{5 \pm \sqrt{361 + 16 y}}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$d = \frac{5 \pm \sqrt{361 + 16 y}}{8}$

Step 5

The final answer is the combination of both solutions.

$d = \frac{5 + \sqrt{361 + 16 y}}{8}$

$d = \frac{5 - \sqrt{361 + 16 y}}{8}$