write the quadratic equation whose roots are -4 and -5 and whose leading coefficient is 5 use the letter x to represent the variable.
use the quadratic formula to solve for x 4x^2-9x+3=0
To write a quadratic equation with roots -4 and -5 and a leading coefficient of 5, we can use the fact that the roots of a quadratic equation are the values of x for which the equation equals zero. So, if the roots are -4 and -5, we can write two quadratic equations as follows:
(x + 4) = 0 --> Equation 1
(x + 5) = 0 --> Equation 2
To find the quadratic equation, we multiply the expressions on the left side of these equations:
(x + 4)(x + 5) = 0
Now, we can multiply these binomials using the distributive property:
x^2 + 5x + 4x + 20 = 0
Combining like terms yields the quadratic equation:
x^2 + 9x + 20 = 0
Therefore, the quadratic equation with roots -4 and -5 and leading coefficient 5 is:
5x^2 + 9x + 20 = 0
Now, let's use the quadratic formula to solve for x in the equation 4x^2 - 9x + 3 = 0:
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the given equation to the standard form (ax^2 + bx + c = 0), we have:
a = 4
b = -9
c = 3
Substituting these values into the quadratic formula, we get:
x = (-(-9) ± √((-9)^2 - 4 * 4 * 3)) / (2 * 4)
x = (9 ± √(81 - 48)) / 8
x = (9 ± √33) / 8
Therefore, the solutions to the equation 4x^2 - 9x + 3 = 0 are:
x = (9 + √33) / 8
x = (9 - √33) / 8