Solve the equation by factoring or by using the quadratic formula: x^2+5x-24=0
To solve the equation x^2 + 5x - 24 = 0, we can use factoring.
We need to find two numbers whose product gives -24 and whose sum gives 5. The numbers that satisfy these conditions are 8 and -3.
So, we can write the equation as (x + 8)(x - 3) = 0.
Setting each factor equal to zero, we have x + 8 = 0 and x - 3 = 0.
Solving these equations, we find x = -8 and x = 3.
Therefore, the solutions to the equation x^2 + 5x - 24 = 0 are x = -8 and x = 3.
To solve the equation x^2 + 5x - 24 = 0, we can solve it by factoring or by using the quadratic formula.
Let's first try to solve it by factoring.
Step 1: Write down the equation.
x^2 + 5x - 24 = 0
Step 2: Start Factoring.
We need to find two numbers that multiply to give -24 and add up to give 5. The numbers that satisfy these conditions are 8 and -3.
(x + 8)(x - 3) = 0
Step 3: Set each factor equal to zero.
x + 8 = 0 or x - 3 = 0
Step 4: Solve for x.
For x + 8 = 0:
x = -8
For x - 3 = 0:
x = 3
Therefore, the solutions to the equation x^2 + 5x - 24 = 0 are x = -8 and x = 3.
Now, let's solve the equation using the quadratic formula.
Step 1: Write down the equation.
x^2 + 5x - 24 = 0
Step 2: Identify the coefficients a, b, and c.
In this equation, a = 1, b = 5, and c = -24.
Step 3: Use the quadratic formula.
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-5 ± √(5^2 - 4(1)(-24))) / (2 * 1)
Simplifying:
x = (-5 ± √(25 + 96)) / 2
x = (-5 ± √121) / 2
x = (-5 ± 11) / 2
So we get two possible solutions:
x = (-5 + 11) / 2 = 6 / 2 = 3
x = (-5 - 11) / 2 = -16 / 2 = -8
Therefore, the solutions to the equation x^2 + 5x - 24 = 0 are x = -8 and x = 3.
To solve the quadratic equation x^2 + 5x - 24 = 0, we can factor it or use the quadratic formula. Let's start with factoring.
Step 1: Write the equation in the form ax^2 + bx + c = 0. In this case, a = 1, b = 5, and c = -24.
Step 2: Factor the quadratic expression. We need to find two numbers whose product is -24 and whose sum is 5.
The numbers that satisfy these conditions are 8 and -3, because 8 * (-3) = -24 and 8 + (-3) = 5.
Step 3: Rewrite the middle term using the two numbers found in Step 2. So, the equation becomes:
x^2 + 8x - 3x - 24 = 0
Step 4: Group the terms and factor out the common factors from each group:
(x^2 + 8x) - (3x + 24) = 0
Step 5: Factor out the greatest common factor from each group:
x(x + 8) - 3(x + 8) = 0
Step 6: Simplify the equation:
(x + 8)(x - 3) = 0
Step 7: Set each factor equal to zero and solve for x:
x + 8 = 0 or x - 3 = 0
For x + 8 = 0, when we solve for x, we get:
x = -8
For x - 3 = 0, when we solve for x, we get:
x = 3
Therefore, the solutions to the equation x^2 + 5x - 24 = 0 are x = -8 and x = 3.
Now, let's solve the equation using the quadratic formula, just to double-check our answer.
The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 5, and c = -24.
Plugging these values into the quadratic formula, we get:
x = (-5 ± √(5^2 - 4(1)(-24))) / (2(1))
Simplifying further:
x = (-5 ± √(25 + 96)) / 2
x = (-5 ± √121) / 2
x = (-5 ± 11) / 2
So, the two possible solutions obtained are:
x = (-5 + 11) / 2 --> x = 6 / 2 --> x = 3
x = (-5 - 11) / 2 --> x = -16 / 2 --> x = -8
We obtain the same solutions as before, x = 3 and x = -8.
Hence, the solutions to the equation x^2 + 5x - 24 = 0 are x = -8 and x = 3.