Problem 11
Question
Solve the quadratic equation by factoring. Check your solutions in the original equation. $$x^{2}-10 x+21=0$$
Step-by-Step Solution
Verified Answer
The solutions to the quadratic equation \(x^{2} - 10x + 21 = 0\) are \(x = 7\) and \(x = 3\).
1Step 1: Factor quadratic equation
First step will be factoring the quadratic equation \(x^{2} - 10x + 21 = 0\). We're looking for two numbers that multiply to 21 (the constant term) and add up to -10 (the coefficient of \(x\)). These numbers are -7 and -3. So, the factoring of our quadratic equation will be: \((x - 7) (x - 3) = 0\).
2Step 2: Solve for \(x\)
Next, set each factor equal to zero and solve for \(x\):\n\(x - 7 = 0 \Rightarrow x = 7\) and \(x - 3 = 0 \Rightarrow x = 3\).
3Step 3: Check the solution
Finally, substitute \(x = 7\) and \(x = 3\) into the original equation to check if it's satisfied: substitute \(x = 7\) into \(x^{2} - 10x + 21 = 0\) getting \(49 - 70 + 21 = 0\), and substitute \(x = 3\) into \(x^{2} - 10x + 21 = 0\) getting \(9 - 30 + 21 = 0\). Both equalities are true, showing our solutions are correct.
Key Concepts
Factoring Quadratic EquationsQuadratic FormulaZero Product PropertyVerification of Solutions
Factoring Quadratic Equations
Solving quadratic equations by factoring is an essential skill in algebra. Factoring involves finding two binomials that when multiplied together give back the original quadratic equation. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The task is to decompose this into a product of two binomials.
For the equation \(x^2 - 10x + 21 = 0\), we look for two numbers that multiply to \(c = 21\) and add up to \(b = -10\). These numbers are -7 and -3. Thus, the factored form becomes \( (x - 7)(x - 3) = 0\). Always ensure the numbers found meet both the multiplication and addition criteria before concluding the factoring process. This method works well when the equation can be factored neatly, often when \(a = 1\) and \(c\) is a product of two integers.
For the equation \(x^2 - 10x + 21 = 0\), we look for two numbers that multiply to \(c = 21\) and add up to \(b = -10\). These numbers are -7 and -3. Thus, the factored form becomes \( (x - 7)(x - 3) = 0\). Always ensure the numbers found meet both the multiplication and addition criteria before concluding the factoring process. This method works well when the equation can be factored neatly, often when \(a = 1\) and \(c\) is a product of two integers.
Quadratic Formula
The quadratic formula provides another method to solve for the roots of a quadratic equation when factoring is not possible or difficult. It is derived from the standard form \(ax^2 + bx + c = 0\) and is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The formula includes the sign \(\pm\), which indicates two solutions, one with addition and one with subtraction. To use the quadratic formula, simply identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation and plug them into the formula. It's especially useful for equations where factoring by inspection is not straightforward, like when the quadratic is not easily decomposable into factors or when the coefficients are large or fractional. The quadratic formula guarantees a solution for any quadratic equation, provided that \(b^2 - 4ac \geq 0\), known as the discriminant.
The formula includes the sign \(\pm\), which indicates two solutions, one with addition and one with subtraction. To use the quadratic formula, simply identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation and plug them into the formula. It's especially useful for equations where factoring by inspection is not straightforward, like when the quadratic is not easily decomposable into factors or when the coefficients are large or fractional. The quadratic formula guarantees a solution for any quadratic equation, provided that \(b^2 - 4ac \geq 0\), known as the discriminant.
Zero Product Property
The Zero Product Property is a critical rule in algebra which states that if a product of factors equals zero, then at least one of the factors must also equal zero.
For the factored equation \( (x - 7)(x - 3) = 0\), we apply this property to find the solutions of the quadratic equation. We set each factor equal to zero: \(x - 7 = 0\) or \(x - 3 = 0\). Solving for \(x\) gives the roots \(x = 7\) and \(x = 3\). The property is used after a quadratic equation has been factored into the product of two binomials. It is a fundamental concept that allows us to go from the factored form back to the individual solutions of the equation.
For the factored equation \( (x - 7)(x - 3) = 0\), we apply this property to find the solutions of the quadratic equation. We set each factor equal to zero: \(x - 7 = 0\) or \(x - 3 = 0\). Solving for \(x\) gives the roots \(x = 7\) and \(x = 3\). The property is used after a quadratic equation has been factored into the product of two binomials. It is a fundamental concept that allows us to go from the factored form back to the individual solutions of the equation.
Verification of Solutions
After solving a quadratic equation, it is crucial to verify that the solutions found are indeed correct. This is done by substituting the solution back into the original equation and checking if it satisfies the equation.
For the example, we check the solutions \(x = 7\) and \(x = 3\) by substituting them into \(x^2 - 10x + 21 = 0\). For \(x = 7\), we get \(49 - 70 + 21 = 0\) which simplifies to \(0 = 0\), confirming that \(x = 7\) is a solution. Similarly, when \(x = 3\), the equation becomes \(9 - 30 + 21 = 0\), which also verifies that \(x = 3\) is a solution. It is a good practice to always perform this verification step to ensure the accuracy of the solutions, avoiding potential calculation mistakes.
For the example, we check the solutions \(x = 7\) and \(x = 3\) by substituting them into \(x^2 - 10x + 21 = 0\). For \(x = 7\), we get \(49 - 70 + 21 = 0\) which simplifies to \(0 = 0\), confirming that \(x = 7\) is a solution. Similarly, when \(x = 3\), the equation becomes \(9 - 30 + 21 = 0\), which also verifies that \(x = 3\) is a solution. It is a good practice to always perform this verification step to ensure the accuracy of the solutions, avoiding potential calculation mistakes.
Other exercises in this chapter
Problem 11
(a) create a scatter plot of the data, (b) draw a line of fit that passes through two of the points, and (c) use the two points to find an equation of the line.
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Determine whether the equation is an identity, a conditional equation, or a contradiction. $$2(x-1)=2 x-2$$
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