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Precalculus Equation Solving and Algebraic Techniques Study Guide

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Q1. Solve the equation: \(2 + \frac{10}{x} = x + 5\)

Background

Topic: Rational Equations

This question tests your ability to solve equations involving rational expressions (fractions with variables in the denominator).

Key Terms and Formulas:

  • Rational equation: An equation containing at least one fraction whose numerator and/or denominator includes a variable.

  • To solve, clear denominators by multiplying both sides by the least common denominator (LCD).

Step-by-Step Guidance

  1. Identify the denominators in the equation. Here, the only denominator is \(x\).

  2. Multiply both sides of the equation by \(x\) to eliminate the fraction.

  3. Distribute and simplify both sides to obtain a quadratic equation in standard form.

  4. Rearrange all terms to one side to set the equation equal to zero.

Try solving on your own before revealing the answer!

Final Answer: x = -5, 2

After simplifying and factoring, you find two possible solutions. Always check for extraneous solutions when solving rational equations.

Q2. Solve the equation: \(\frac{a}{a-6} = \frac{-5}{a-4}\)

Background

Topic: Rational Equations

This question tests your ability to solve equations with variables in the denominators on both sides.

Key Terms and Formulas:

  • Cross-multiplication: Used when you have a proportion (two fractions set equal).

Step-by-Step Guidance

  1. Cross-multiply to eliminate the denominators: \(a(a-4) = -5(a-6)\).

  2. Expand both sides to get a quadratic equation.

  3. Move all terms to one side to set the equation to zero.

  4. Factor or use the quadratic formula to solve for \(a\).

Try solving on your own before revealing the answer!

Final Answer: a = -6, 5

After solving, check for extraneous solutions by substituting back into the original denominators.

Q3. Solve the equation: \(\sqrt{6x + 6} = 6\)

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots.

Key Terms and Formulas:

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

  • To solve, isolate the radical and then square both sides to eliminate the square root.

Step-by-Step Guidance

  1. Isolate the square root on one side (already done here).

  2. Square both sides to remove the square root: \((\sqrt{6x + 6})^2 = 6^2\).

  3. Solve the resulting linear equation for \(x\).

Try solving on your own before revealing the answer!

Final Answer: x = 5

Always check your solution in the original equation to ensure it does not produce an extraneous result.

Q4. Solve the radical equation: \(\sqrt{7 - 21x} = x + 9\)

Background

Topic: Radical Equations

This question tests your ability to solve equations where the variable appears both inside and outside a square root.

Key Terms and Formulas:

  • Isolate the radical, then square both sides to eliminate the square root.

Step-by-Step Guidance

  1. Isolate the square root (already done).

  2. Square both sides: \((\sqrt{7 - 21x})^2 = (x + 9)^2\).

  3. Expand the right side and rearrange to form a quadratic equation.

  4. Solve the quadratic equation for \(x\).

Try solving on your own before revealing the answer!

Final Answer: x = -2

Check your solution in the original equation to avoid extraneous roots.

Q5. Decide what number must be added to each binomial to make a perfect square trinomial: \(x^2 + 12x\)

Background

Topic: Completing the Square

This question tests your understanding of how to create a perfect square trinomial from a binomial.

Key Terms and Formulas:

  • Perfect square trinomial: An expression of the form \((x + a)^2 = x^2 + 2ax + a^2\).

  • To complete the square, add \(\left(\frac{b}{2}\right)^2\) to \(x^2 + bx\).

Step-by-Step Guidance

  1. Identify \(b\) in \(x^2 + bx\). Here, \(b = 12\).

  2. Calculate \(\left(\frac{12}{2}\right)^2\).

  3. Add this value to the binomial to form the perfect square trinomial.

Try solving on your own before revealing the answer!

Final Answer: x^2 + 12x + 36

The number to add is 36, making the trinomial a perfect square.

Q6. Decide what number must be added to the binomial to make a perfect square trinomial: \(x^2 - 8x\)

Background

Topic: Completing the Square

This question tests your ability to complete the square for a binomial with a negative coefficient.

Key Terms and Formulas:

  • Use \(\left(\frac{b}{2}\right)^2\) where \(b\) is the coefficient of \(x\).

Step-by-Step Guidance

  1. Identify \(b = -8\).

  2. Calculate \(\left(\frac{-8}{2}\right)^2\).

  3. Add this value to the binomial to form the perfect square trinomial.

Try solving on your own before revealing the answer!

Final Answer: x^2 - 8x + 16

The number to add is 16, making the trinomial a perfect square.

Q7. Solve the equation using the quadratic formula: \(x^2 + 6x - 4 = 0\)

Background

Topic: Quadratic Equations

This question tests your ability to use the quadratic formula to solve a quadratic equation.

Key Terms and Formulas:

  • Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

  • For \(ax^2 + bx + c = 0\), identify \(a\), \(b\), and \(c\).

Step-by-Step Guidance

  1. Identify \(a = 1\), \(b = 6\), \(c = -4\).

  2. Plug these values into the quadratic formula.

  3. Calculate the discriminant \(b^2 - 4ac\).

  4. Set up the expression for \(x\) using the quadratic formula, but do not simplify fully yet.

Try solving on your own before revealing the answer!

Final Answer: x = -3 \pm \sqrt{13}

These are the exact solutions using the quadratic formula.

Q8. Solve the following quadratic equation using the quadratic formula: \(2x^2 - 2x + 7 = 0\)

Background

Topic: Quadratic Equations with Complex Solutions

This question tests your ability to use the quadratic formula and handle complex numbers.

Key Terms and Formulas:

  • Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

  • Complex solutions occur when the discriminant is negative.

Step-by-Step Guidance

  1. Identify \(a = 2\), \(b = -2\), \(c = 7\).

  2. Plug these values into the quadratic formula.

  3. Calculate the discriminant \((-2)^2 - 4 \cdot 2 \cdot 7\).

  4. Set up the expression for \(x\) using the quadratic formula, but do not simplify fully yet.

Try solving on your own before revealing the answer!

Final Answer: x = \frac{1 \pm i\sqrt{13}}{2}

The solutions are complex numbers because the discriminant is negative.

Q9. Multiply both sides of the equation \(\frac{7x}{3} + 4 = \frac{1}{6}\) by 6.

Background

Topic: Clearing Fractions in Equations

This question tests your ability to eliminate fractions by multiplying both sides by the least common denominator (LCD).

Key Terms and Formulas:

  • LCD: The smallest number that is a multiple of all denominators in the equation.

Step-by-Step Guidance

  1. Identify the denominators: 3 and 6. The LCD is 6.

  2. Multiply every term on both sides of the equation by 6.

  3. Simplify each term after multiplying to clear the fractions.

Try solving on your own before revealing the answer!

Final Answer: 14x + 24 = 1

All fractions are cleared, making the equation easier to solve.

Q10. Solve the equation: \(\frac{6y}{y-8} + 2 = \frac{4y}{y-8}\)

Background

Topic: Rational Equations

This question tests your ability to solve equations with rational expressions and a common denominator.

Key Terms and Formulas:

  • Combine like terms and isolate the variable.

Step-by-Step Guidance

  1. Subtract \(\frac{4y}{y-8}\) from both sides to combine like terms.

  2. Simplify the equation to isolate \(y\).

  3. Solve for \(y\) by isolating the variable.

Try solving on your own before revealing the answer!

Final Answer: y = 4

Check your solution to ensure it does not make any denominator zero.

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