Practice Makes Perfect Algebra - Carolyn Wheater [21]
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Quadratic equations and their graphs
Quadratic equations are equations that contain a term in which the variable is squared. The standard form of a quadratic equation is ax2 + bx + c = 0, but you may have to do some rearranging to get the equation you’re given into that form. The x-term or the constant term may be missing if b equals 0 or c equals 0, but if the x2 term is missing, it’s not a quadratic equation. The graph of a quadratic equation y = ax2 + bx + c has a particular shape called a parabola.
Solving by square roots
If a quadratic equation contains just a squared term and a constant term, you can solve it by moving the terms to opposite sides of the equal sign and taking the square root of both sides. Remember that there is both a positive and a negative square root of any positive number.
If the constant is not a perfect square, leave solutions in simplest radical form, unless there’s a very good reason to use a decimal approximation.
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EXERCISE 9.1
Solve each equation by isolating the square and taking the square root of both sides.
1. x2 = 64
2. x2 − 16 = 0
3. x2 − 8 = 17
4. x2 = 18
5. 3x2 = 48
6. t2 − 1000 = 0
7. 2y2 − 150 = 0
8. 9x2 = 4
9. 64y2 = 25
10. 4x2 − 15 = 93
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Completing the square
If the quadratic equation has just a square term and a constant term, you can solve it by taking the square root of both sides. If it has three terms that happen to form a perfect square trinomial, you can rewrite it as the square of a binomial and then solve by taking the square root of both sides. Much of the time, however, the polynomial is not a perfect square.
Completing the square is a process that turns one side of the equation into a perfect square trinomial so that you can solve by taking the square root of both sides. Of course, completing the square doesn’t just magically change one side of the equation. It adds the same number to both sides so that the new equation is equivalent to the original. The key is to know what to add.
To complete the square, move the constant to the opposite side of the equation from the x2 and x terms.
3x2 + 24 x + 15 = 0
3x2 + 24 x = −15
Divide both sides of the equation by the coefficient of x2.
3x2 + 24 x = −15
x2 + 8 x = −5
Take half the coefficient of x, square it, and add the result to both sides.
x2 + 8x = −5
x2 + 8x + 42 = −5 + 42
x2 + 8 x + 16 = −5 + 16
(x + 4)2 = 11
Solve the equation by taking the square root of both sides and then isolating the variable.
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EXERCISE 9.2
Solve each equation. Do not FOIL the binomial square. Take the square root of both sides. Give answers in simplest radical form.
1.(x − 2)2 = 25
2.(x + 1)2 = 9
3.(x − 3)2 = 48
4.(x + 1)2 = 75
5.(3x − 5)2 = 12
Complete the square and solve the equation.
6. y2 − 8y − 7 = 0
7. x2 − 5x = 14
8. x2 + 4x − 4 = 0
9. a2 + 5a − 3 = 0
10. t2 = 10t — 8
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The quadratic formula
Completing the square is a very effective method for solving quadratic equations, but it can get complicated and the numbers can get messy, so you soon find yourself wishing for an easier way. The quadratic formula is a shortcut to the solution you would have obtained by completing the square. Using it is a bit complicated, but easier than doing all the work of completing the square.
If ax2 + bx + c = 0, then . You just pick the values of a, b, and c out of the equation, plug them into the formula, and simplify. Be certain your equation is in ax2 + bx + c = 0 form before deciding on the values of a, b, and c.
To solve 5x = 2 − 3x2, first put the equation in standard form. Compare 3x2 + 5x − 2 = 0 to ax2 + bx + c = 0, and you find that a = 3, b = 5, and c = −2. Plug those values into the formula.
Follow the order of operations and watch your signs as you simplify.
The two solutions that are typical of quadratic equations come from that ± sign.
The discriminant
One portion of the quadratic formula, called the discriminant, can give you useful information about the number and type of solutions