4.10: Finding all Real Roots of a Function
To find the real roots of a function, find where the function intersects the x-axis. To find where the function intersects the x-axis, set \(f(x) = 0\) and solve the equation for \(x\).
If the function is a linear function of degree 1, \(f(x) = mx + b\) and the x-intercept is the root of the equation, found by solving the equation for \(x\). To find the roots of quadratic equations, there are several ways to find the zeros:
- Fully factor the quadratic expression.
- Use the quadratic formula, with the quadratic equation in the form \(Ax^2 + Bx + C = 0\).
- Complete the square on the quadratic expression (not included in this workbook).
Some cubic equations can also be solved easily, if the polynomial can be factored to find the zeroes. Also, the cubic equation can be factored if written in the form of a sum or difference of perfect cubes. If they are not in this form, then a calculator or a computer could find the roots of a cubic equation.
The focus of our class is to work with polynomials whose roots can be found using traditional algebraic techniques. For details about how to factor an expression, please refer to the section Factoring/Finding Polynomial Solutions (zeros). For details about how to use the Quadratic Formula, please refer to that section in the document.
Find the real roots of each equation by factoring or using the Quadratic Formula. Express the exact final simplified answers (real numbers or simplified radical expressions).
- \(x ^2 + x − 12 = 0\)
- \(−6x ^2 + x + 12 = 0\)
- \(4x ^2 + 5x − 6 = 0\)
- \(\dfrac{1 }{2} a^2 + a − 12 = 0\)
- \(2x^2 + 7x − 15=0\)
- \(12x^2 − 9x − 3 = 0\)