Eighth Grade Mathematics: Solving Equations of the Form x^2 = p and x^3 = p
Equations involving variables raised to the second or third power appear across physical science and geometry, from calculating the stopping distance of an automobile to designing structural components and spherical containers. Solving equations of the form x^2 = p and x^3 = p requires applying inverse operations to isolate the variable. In this chapter, you will learn to solve quadratic and cubic power equations, navigate the dual solutions of square root equations, and classify whether solutions are rational or irrational.
Solving Quadratic Equations of the Form x^2 = p
A quadratic equation in the form x^2 = p states that an unknown quantity x, when multiplied by itself, equals the constant value p. To isolate x, you apply the inverse operation of squaring, which is taking the square root of both sides.
Solving x^2 = p:
If p > 0, x = +-sqrt(p) (Two real solutions: one positive, one negative)
If p = 0, x = 0 (One real solution)
If p < 0, No real solution (A real number squared cannot be negative)
The plus-or-minus symbol (+-) is mandatory whenever solving x^2 = p for a positive constant p. Many students forget the negative solution, but both positive and negative values satisfy the equation.
Example 1: Solve x^2 = 64
x = +-sqrt(64)
x = +-8 (Two solutions: x = 8 and x = -8)
Check: (8)^2 = 64 and (-8)^2 = 64. Both are valid!
Rational versus Irrational Solutions
When solving x^2 = p, the nature of the solution depends on whether p is a perfect square.
Case A: p is a perfect square (Rational solutions)
x^2 = 49 / 81 ---> x = +-sqrt(49 / 81) = +- 7/9 (Rational)
Case B: p is not a perfect square (Irrational solutions)
x^2 = 18 ---> x = +-sqrt(18) (Irrational)
When p is not a perfect square, the exact solutions are written in radical form as +-sqrt(p). If a problem asks for an approximation, you can evaluate the radical using bounding methods or a calculator (for instance, +-sqrt(18) is approximately +-4.24). Unless instructed otherwise, always leave the answer in exact radical form.
Solving Cubic Equations of the Form x^3 = p
A cubic equation in the form x^3 = p states that an unknown quantity x, multiplied by itself three times, equals p. To isolate x, apply the inverse operation of cubing, which is taking the cube root of both sides.
Solving x^3 = p:
For ANY real number p (positive, zero, or negative),
there is ALWAYS exactly ONE real solution:
x = cbrt(p)
Unlike quadratic equations, cubic equations do NOT use the plus-or-minus symbol. A positive number cubed yields a positive number, and a negative number cubed yields a negative number.
Example 1: Solve x^3 = 125
x = cbrt(125) = 5 (Only ONE solution: 5)
Example 2: Solve x^3 = -27
x = cbrt(-27) = -3 (Only ONE solution: -3)
Example 3: Solve x^3 = 50
x = cbrt(50) (Exact irrational solution)
Cubing is a one-to-one function across the real numbers; every real number has one and only one real cube root.
Multi-Step Equations Leading to x^2 = p or x^3 = p
Before extracting a root, the variable term containing the exponent must be completely isolated on one side of the equation using standard inverse operations.
Solve: 3*x^2 - 7 = 41
Step 1: Add 7 to both sides:
3*x^2 = 48
Step 2: Divide both sides by 3 to isolate x^2:
x^2 = 16
Step 3: Take the square root of both sides (include +-):
x = +-sqrt(16)
x = +-4
Attempting to take a square root before isolating the squared term leads to severe algebraic errors. Always isolate the squared or cubed variable completely before extracting roots.
Chapter Exercises
Exercise 1: Solve the equation 5*x^2 + 12 = 92. State whether the solutions are rational or irrational, and verify both solutions in the original equation.
Exercise 2: Solve the cubic equation 2*x^3 - 9 = 45. Find the exact solution, state whether it is rational or irrational, and verify it in the original equation.
Exercise 3: A student solves the equation x^2 = 20 and writes down x = 4.47. A second student writes down x = +-sqrt(20). Compare these two answers in terms of mathematical exactness and completeness, explaining which answer is superior and why.
Solutions and Explanations
Solution 1: To solve 5x^2 + 12 = 92, begin by isolating the x^2 term. Subtract 12 from both sides of the equation to get 5x^2 = 80. Next, divide both sides by 5, which gives x^2 = 16. Finally, take the square root of both sides, remembering to include both positive and negative roots: x = +-sqrt(16), so x = 4 or x = -4. Because 4 and -4 can be written as 4/1 and -4/1, these solutions are rational numbers. To verify x = 4, substitute it into the original equation: 5(4)^2 + 12 = 5(16) + 12 = 80 + 12 = 92. To verify x = -4: 5(-4)^2 + 12 = 5(16) + 12 = 80 + 12 = 92. Both solutions satisfy the equation perfectly.
Solution 2: To solve 2x^3 - 9 = 45, begin by isolating the x^3 term. Add 9 to both sides of the equation to obtain 2x^3 = 54. Next, divide both sides by 2 to yield x^3 = 27. Now, take the cube root of both sides: x = cbrt(27). Since 3 * 3 * 3 = 27, the exact solution is x = 3. There is no negative solution because (-3)^3 = -27. The number 3 is an integer, so the solution is a rational number. To verify, substitute x = 3 into the original equation: 2(3)^3 - 9 = 2(27) - 9 = 54 - 9 = 45, confirming the solution is correct.
Solution 3: The second student's answer, x = +-sqrt(20), is vastly superior in both completeness and mathematical exactness. First, regarding completeness, the first student provided only the positive value 4.47, completely missing the negative solution -4.47; any quadratic equation of the form x^2 = p with positive p must possess two real solutions. Second, regarding exactness, 20 is not a perfect square, which means its square root is an irrational number whose decimal expansion continues forever without repeating. The value 4.47 is merely a rounded approximation (4.47 squared is 19.9809, not 20). Writing x = +-sqrt(20) represents the exact, unrounded mathematical values while capturing both the positive and negative roots.