SOLUTION BY THEQUADRATIC FORMULA
The quadratic formula is derived by applying the process of completing the square tosolve for x in the general form of the quadratic equation, ax2 + bx + c = 0. Remember that the general form represents every possible quadratic equation. Thus, if we can solve this equation for x, the solution will be in terms of a, b, and c. To solve this equation for x by completing the square, we proceed as follows:
1. Subtract the constant term, c, from bothmembers.
ax2 + bx + c = 0
2. Divide all terms by a so that the coefficient of the x2 term becomes unity.
3. Add the square of one-half the coefficientof the x term b/a, to both members.
4. Factor the left member and simplify theright member.
5. Take the square root of both members.
6. Solve for x.
Thus, we have solved the equation representing every quadratic for its unknown in termsof its constants a, b, and c. Hence, in a given quadratic we need only substitute in the expression
the values of a, b, and c, as they appear in theparticular equation, to derive the roots of that equation. This expression is called the QUADRATIC FORMULA. The gene r a 1 quadratic equation, ax2 + bx + c = 0, and the quadratic formula should be memorized. Then, when a quadratic cannot be solved quickly by factoring, it may be solved at once by the formula.
EXAMPLE: Use the quadratic formula to solvethe equation
x2 + 30 - 11x = 0.
1. Set up the equation in standard form.
X2- 11x + 30 = 0
Then a (coefficient of x2) = 1
c (the constant term) = 30
EXAMPLE: Find the roots of
Substituting into the quadratic formula gives
The two roots are
These roots are irrational numbers, since the radicals cannot be removed.
If the decimal values of the roots are desired, the value of the square root of 17 can be taken from appendix I of this course. Substituting = 4.1231 and simplifying gives
In decimal form, the roots of 2x2 - 3x -1 = 0 to the nearest tenth are 1.8 and -0.3.
Notice that the subscripts, 1 and 2, are usedto distinguish between the two roots of the equation. The three roots of a cubic equation in x might be designated x1, x2, and x1. Sometimes the letter r is used for root. Using r, the roots of a cubic equation could be labeled r1, r2, and r3.
Multiplying both members of the equation by 8,the LCD, we have
Practice problems. Use the quadratic formula to find the roots of the following equations: