Solving Quadratic and Higher-Degree Equations: Methods and Examples

Solving Quadratic and Higher-Degree Equations

Completing the Square

Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. This technique rewrites the quadratic in a perfect square form, making it easier to solve for $x$.

• Step 1: If the leading coefficient $a$ is not 1, divide both sides of the equation by $a$.

• Step 2: Isolate the variable terms on one side and the constant term on the other.

• Step 3: Half the coefficient of $x$, square the result, and add it to both sides.

• Step 4: Factor the left side as a perfect square trinomial.

Example:

• Given: $9x^2 - 6x - 4 = 0$

• Divide by 9: $x^2 - \frac{2}{3}x - \frac{4}{9} = 0$

• Isolate variable terms: $x^2 - \frac{2}{3}x = \frac{4}{9}$

• Half of $-\frac{2}{3}$ is $-\frac{1}{3}$; square it: $\left(-\frac{1}{3}\right)^2 = \frac{1}{9}$

• Add $\frac{1}{9}$ to both sides: $x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{4}{9} + \frac{1}{9}$

• Factor: $(x - \frac{1}{3})^2 = \frac{5}{9}$

• Solve: $x - \frac{1}{3} = \pm \frac{\sqrt{5}}{3}$

• Final answer: $x = \frac{1}{3} \pm \frac{\sqrt{5}}{3}$

Factoring Quadratic Equations

Factoring is another method to solve quadratic equations by expressing the quadratic as a product of two binomials and setting each factor to zero.

• Example: $x^2 - 6x + 4 = 0$ (not easily factorable; use quadratic formula or completing the square)

• Example: $2x^2 - 7x + 3 = 0$

• Factor: $(2x - 1)(x - 3) = 0$

• Solutions: $x = \frac{1}{2},\ 3$

Solving Higher-Degree Polynomial Equations

Equations of degree higher than two can often be solved by factoring, substitution, or by reducing them to quadratic form.

1. Factoring by Grouping

• Example: $x^3 + x^2 - 4x - 4 = 0$

• Group: $(x^3 + x^2) + (-4x - 4) = 0$

• Factor: $x^2(x + 1) - 4(x + 1) = 0$

• $(x + 1)(x^2 - 4) = 0$

• Further factor: $(x + 1)(x - 2)(x + 2) = 0$

• Solutions: $x = -1, 2, -2$

2. Substitution (Quadratic in Disguise)

• Example: $x^4 - 8x^2 - 9 = 0$

• Let $v = x^2$; equation becomes $v^2 - 8v - 9 = 0$

• Factor: $(v - 9)(v + 1) = 0$

• So $v = 9$ or $v = -1$

• Back-substitute: $x^2 = 9 \implies x = \pm 3$; $x^2 = -1 \implies x = \pm i$

3. Solving Radical Equations

To solve equations involving radicals, isolate the radical and then square both sides to eliminate it. Check for extraneous solutions.

• Example: $\sqrt{2x+1} + 2 = x$

• Isolate: $\sqrt{2x+1} = x - 2$

• Square both sides: $2x + 1 = (x - 2)^2$

• Expand: $2x + 1 = x^2 - 4x + 4$

• Rearrange: $0 = x^2 - 6x + 3$

• Use quadratic formula: $x = \frac{6 \pm \sqrt{36 - 12}}{2} = \frac{6 \pm \sqrt{24}}{2} = \frac{6 \pm 2\sqrt{6}}{2} = 3 \pm \sqrt{6}$

• Check for extraneous solutions by substituting back into the original equation.

4. Solving Equations with Rational Exponents

• Example: $x^{2/3} - 8 = -4$

• Add 8: $x^{2/3} = 4$

• Raise both sides to the $3/2$ power: $x = 4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$

• So $x = 8$

Quadratic Formula

The quadratic formula provides the solution to any quadratic equation $ax^2 + bx + c = 0$:

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• The discriminant $b^2 - 4ac$ determines the nature of the roots:

  • If $b^2 - 4ac > 0$, two real solutions

  • If $b^2 - 4ac = 0$, one real solution

  • If $b^2 - 4ac < 0$, two complex solutions

Summary Table: Methods for Solving Equations

Equation Type	Recommended Method	Example
Quadratic ($ax^2 + bx + c = 0$)	Factoring, Completing the Square, Quadratic Formula	$x^2 - 6x + 4 = 0$
Higher-Degree Polynomial	Factoring, Substitution, Grouping	$x^3 + x^2 - 4x - 4 = 0$
Radical Equations	Isolate Radical, Square Both Sides	$\sqrt{2x+1} + 2 = x$
Rational Exponents	Isolate Power, Raise Both Sides to Reciprocal Exponent	$x^{2/3} - 8 = -4$

Key Definitions

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.

• Radical Equation: An equation in which the variable is under a root.

• Rational Exponent: An exponent that is a fraction, such as $x^{m/n}$.

• Extraneous Solution: A solution that emerges from the process of solving but does not satisfy the original equation.

Additional info:

• Always check for extraneous solutions when solving radical equations or equations with rational exponents.

• Factoring by grouping is especially useful for cubic and quartic polynomials.