In the Quadratic Formula, the expression underneath the radical symbol determines the number and type of solutions the formula will reveal. These examples have shown that a quadratic equation may have two real solutions, one real solution, or two complex solutions. The following example is a little different. Most of the quadratic equations you've looked at have two solutions, like the one above. The power of the Quadratic Formula is that it can be used to solve any quadratic equation, even those where finding number combinations will not work. Sometimes, it may be easier to solve an equation using conventional factoring methods, like finding number pairs that sum to one number (in this example, 4) and that produce a specific product (in this example −5) when multiplied. However, upon looking at x 2 + 4 x = 5, you may have thought “I already know how to do this I can rewrite this equation as x 2 + 4 x – 5 = 0, and then factor it as ( x + 5)( x – 1) = 0, so x = −5 or 1.” This is correct-and congratulations if you made this connection! You’ve solved the equation successfully using the Quadratic Formula! You get two true statements, so you know that both solutions work: x = 1 or −5. This means the correct answer is a = 1, b = 3, and c = −6. Putting the terms in order gives the standard form x 2 + 3 x – 6 = 0. You correctly found that 3 x + x 2 = 6 becomes 3 x + x 2 – 6 = 0. Remember that in standard form, the equation is written in the form ax 2 + bx + c = 0. 3 x + x 2 = 6 becomes 3 x + x 2 – 6 = 0, so the standard form is x 2 + 3 x – 6 = 0. This means the correct answer is a = 1, b = 3, and c = −6.Ĭorrect. The c must be on the left side of the equation. You put the terms in the correct order, but the right side must be equal to 0. ![]() Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.Incorrect. This is demonstrated by the graph provided below. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. ![]() ![]() Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. Below is the quadratic formula, as well as its derivation.įrom this point, it is possible to complete the square using the relationship that:Ĭontinuing the derivation using this relationship: Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. For example, a cannot be 0, or the equation would be linear rather than quadratic. The numerals a, b, and c are coefficients of the equation, and they represent known numbers. Where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. In algebra, a quadratic equation is any polynomial equation of the second degree with the following form: ![]() Fractional values such as 3/4 can be used.
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