y^2+5y-24-এর সমাধান করুন

$$y ^ { 2 } + 5 y - 24$$

Factor

$$\left(y-3\right)\left(y+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-24. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=-3 b=8$$

Rewrite y^{2}+5y-24 as \left(y^{2}-3y\right)+\left(8y-24\right).

$$\left(y^{2}-3y\right)+\left(8y-24\right)$$

Factor out y in the first and 8 in the second group.

$$y\left(y-3\right)+8\left(y-3\right)$$

Factor out common term y-3 by using distributive property.

$$\left(y-3\right)\left(y+8\right)$$

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.

$$y^{2}+5y-24=0$$

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.

$$y=\frac{-5±\sqrt{5^{2}-4\left(-24\right)}}{2}$$

Square 5.

$$y=\frac{-5±\sqrt{25-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$y=\frac{-5±\sqrt{25+96}}{2}$$

Add 25 to 96.

$$y=\frac{-5±\sqrt{121}}{2}$$

Take the square root of 121.

$$y=\frac{-5±11}{2}$$

Now solve the equation y=\frac{-5±11}{2} when ± is plus. Add -5 to 11.

$$y=\frac{6}{2}$$

Divide 6 by 2.

$$y=3$$

Now solve the equation y=\frac{-5±11}{2} when ± is minus. Subtract 11 from -5.

$$y=-\frac{16}{2}$$

Divide -16 by 2.

$$y=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and -8 for x_{2}.

$$y^{2}+5y-24=\left(y-3\right)\left(y-\left(-8\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$y^{2}+5y-24=\left(y-3\right)\left(y+8\right)$$

Evaluate

$$\left(y-3\right)\left(y+8\right)$$