Solve 4j^2=-7+29j

$$4 j ^ { 2 } = - 7 + 29 j$$

Solve for j

$$j=\frac{1}{4}=0.25$$

$$j=7$$

Steps Using Factoring By Grouping

Subtract -7 from both sides.

$$4j^{2}-\left(-7\right)=29j$$

The opposite of -7 is 7.

$$4j^{2}+7=29j$$

Subtract 29j from both sides.

$$4j^{2}+7-29j=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$4j^{2}-29j+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4j^{2}+aj+bj+7. To find a and b, set up a system to be solved.

$$a+b=-29 ab=4\times 7=28$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 28.

$$-1,-28 -2,-14 -4,-7$$

Calculate the sum for each pair.

$$-1-28=-29 -2-14=-16 -4-7=-11$$

The solution is the pair that gives sum -29.

$$a=-28 b=-1$$

Rewrite 4j^{2}-29j+7 as \left(4j^{2}-28j\right)+\left(-j+7\right).

$$\left(4j^{2}-28j\right)+\left(-j+7\right)$$

Factor out 4j in the first and -1 in the second group.

$$4j\left(j-7\right)-\left(j-7\right)$$

Factor out common term j-7 by using distributive property.

$$\left(j-7\right)\left(4j-1\right)$$

To find equation solutions, solve j-7=0 and 4j-1=0.

$$j=7 j=\frac{1}{4}$$

Steps Using the Quadratic Formula

Subtract -7 from both sides.

$$4j^{2}-\left(-7\right)=29j$$

The opposite of -7 is 7.

$$4j^{2}+7=29j$$

Subtract 29j from both sides.

$$4j^{2}+7-29j=0$$

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.

$$4j^{2}-29j+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -29 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$j=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Square -29.

$$j=\frac{-\left(-29\right)±\sqrt{841-4\times 4\times 7}}{2\times 4}$$

Multiply -4 times 4.

$$j=\frac{-\left(-29\right)±\sqrt{841-16\times 7}}{2\times 4}$$

Multiply -16 times 7.

$$j=\frac{-\left(-29\right)±\sqrt{841-112}}{2\times 4}$$

Add 841 to -112.

$$j=\frac{-\left(-29\right)±\sqrt{729}}{2\times 4}$$

Take the square root of 729.

$$j=\frac{-\left(-29\right)±27}{2\times 4}$$

The opposite of -29 is 29.

$$j=\frac{29±27}{2\times 4}$$

Multiply 2 times 4.

$$j=\frac{29±27}{8}$$

Now solve the equation j=\frac{29±27}{8} when ± is plus. Add 29 to 27.

$$j=\frac{56}{8}$$

Divide 56 by 8.

$$j=7$$

Now solve the equation j=\frac{29±27}{8} when ± is minus. Subtract 27 from 29.

$$j=\frac{2}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$j=\frac{1}{4}$$

The equation is now solved.

$$j=7 j=\frac{1}{4}$$

Steps for Completing the Square

Subtract 29j from both sides.

$$4j^{2}-29j=-7$$

Divide both sides by 4.

$$\frac{4j^{2}-29j}{4}=-\frac{7}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$j^{2}-\frac{29}{4}j=-\frac{7}{4}$$

Divide -\frac{29}{4}, the coefficient of the x term, by 2 to get -\frac{29}{8}. Then add the square of -\frac{29}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$j^{2}-\frac{29}{4}j+\left(-\frac{29}{8}\right)^{2}=-\frac{7}{4}+\left(-\frac{29}{8}\right)^{2}$$

Square -\frac{29}{8} by squaring both the numerator and the denominator of the fraction.

$$j^{2}-\frac{29}{4}j+\frac{841}{64}=-\frac{7}{4}+\frac{841}{64}$$

Add -\frac{7}{4} to \frac{841}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$j^{2}-\frac{29}{4}j+\frac{841}{64}=\frac{729}{64}$$

Factor j^{2}-\frac{29}{4}j+\frac{841}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(j-\frac{29}{8}\right)^{2}=\frac{729}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(j-\frac{29}{8}\right)^{2}}=\sqrt{\frac{729}{64}}$$

Simplify.

$$j-\frac{29}{8}=\frac{27}{8} j-\frac{29}{8}=-\frac{27}{8}$$

Add \frac{29}{8} to both sides of the equation.

$$j=7 j=\frac{1}{4}$$