Problem 53
Question
Solve the equation. $$x^{2}-6 x+7=0$$
Step-by-Step Solution
Verified Answer
The roots of the equation are \(x = 3 + \sqrt{2}\) and \(x = 3 - \sqrt{2}\).
1Step 1: Identify a, b and c
From the equation, we have \(a = 1\), \(b = -6\), and \(c = 7\). The quadratic formula requires these constants to be plugged in to find the roots of the equation.
2Step 2: Substitute into the quadratic formula
Substitute the values of a, b and c into the quadratic formula. So, \(x = [-(-6) ± \sqrt{(-6)^2 - 4*1*7}] / (2*1)\).
3Step 3: Simplify the expression
Before simplifying, the expression becomes: \(x = [6 ± \sqrt{36 - 28}] / 2\), which simplifies to: \(x = [6 ± \sqrt{8}] / 2\).
4Step 4: Resolve the radical
Next, resolve the radical. The expression inside the square root, 8, is a perfect square number, so it can be simplified: \(x = [6 ± 2\sqrt{2}] / 2\).
5Step 5: Simplify the equation
Finally, simplify the entire equation to find the solution: \(x = 3 ± \sqrt{2}\). Hence the roots of the equation are \(x = 3 + \sqrt{2}\) and \(x = 3 - \sqrt{2}\).
Key Concepts
Quadratic FormulaFactoring QuadraticsRoots of PolynomialRadical Expression
Quadratic Formula
The quadratic formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \).
To use this formula, you need to identify the coefficients (a, b, and c) from the quadratic equation. Once determined, these coefficients are inserted into the formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This provides the solutions, or roots, of the quadratic equation. In our exercise, we identified \( a = 1 \) , \( b = -6 \) and \( c = 7 \), plugged them into the formula, and found our roots.
Understanding the quadratic formula is key because it always works for any quadratic equation, whether or not it can be factored easily, making it a versatile and powerful tool for students.
To use this formula, you need to identify the coefficients (a, b, and c) from the quadratic equation. Once determined, these coefficients are inserted into the formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This provides the solutions, or roots, of the quadratic equation. In our exercise, we identified \( a = 1 \) , \( b = -6 \) and \( c = 7 \), plugged them into the formula, and found our roots.
Understanding the quadratic formula is key because it always works for any quadratic equation, whether or not it can be factored easily, making it a versatile and powerful tool for students.
Factoring Quadratics
Another method for solving quadratics is factoring. Factoring converts the quadratic equation into a product of binomials. We look for two numbers that multiply to give 'ac' and add up to give 'b'.
However, in the given problem \( x^2 - 6x + 7 = 0 \), factoring is not easily applicable. To successfully factor a quadratic, the middle term, here represented by '-6x', should ideally be split into two terms whose coefficients multiply to the constant term 'c', which is 7 in our case. In some scenarios when quadratics cannot be factored using integers or when factoring is complicated, it's usually more practical to use the quadratic formula.
However, in the given problem \( x^2 - 6x + 7 = 0 \), factoring is not easily applicable. To successfully factor a quadratic, the middle term, here represented by '-6x', should ideally be split into two terms whose coefficients multiply to the constant term 'c', which is 7 in our case. In some scenarios when quadratics cannot be factored using integers or when factoring is complicated, it's usually more practical to use the quadratic formula.
Roots of Polynomial
The solutions to a polynomial equation, also known as the roots, are the values of 'x' at which the polynomial equals zero. In a quadratic equation, there can be up to two real roots.
For the quadratic in question, we used the quadratic formula to determine its roots. It's important to note that roots are the 'x-intercepts' or 'zeros' of the quadratic function when graphed. The discriminant, \( b^2 - 4ac \), within the quadratic formula indicates the nature of the roots. If the discriminant is positive, there are two distinct real roots, as in our exercise.
For the quadratic in question, we used the quadratic formula to determine its roots. It's important to note that roots are the 'x-intercepts' or 'zeros' of the quadratic function when graphed. The discriminant, \( b^2 - 4ac \), within the quadratic formula indicates the nature of the roots. If the discriminant is positive, there are two distinct real roots, as in our exercise.
Radical Expression
A radical expression contains a square root, cube root, or higher-order root symbol. In the quadratic formula, the expression under the square root symbol is called the 'radicand'. Simplifying the radical expression is crucial to solving the quadratic formula.
In the example \( x = [6 \pm \sqrt{36 - 28}] / 2 \), the radical expression \( \sqrt{8} \) was simplified to \( \sqrt{4 \cdot 2} \) or \( 2\sqrt{2} \). It’s always simpler working with small numbers inside a square root, which is why simplifying the radical, when possible, is a good practice.
In the example \( x = [6 \pm \sqrt{36 - 28}] / 2 \), the radical expression \( \sqrt{8} \) was simplified to \( \sqrt{4 \cdot 2} \) or \( 2\sqrt{2} \). It’s always simpler working with small numbers inside a square root, which is why simplifying the radical, when possible, is a good practice.
Other exercises in this chapter
Problem 53
A library has received a single large contribution of \(\$ 5000 .\) A walkathon is also being held in which each sponsor will contribute \(\$ 10 .\) Let \(x\) r
View solution Problem 53
The variables \(x\) and \(y\) vary directly. Use the given values of the variables to write an equation that relates \(x\) and \(y .\) $$x=4, y=8$$
View solution Problem 54
Sketch the graph of the function. $$y=5 x^{2}+4 x-5$$
View solution Problem 54
Find all square roots of the number or write no square roots. Check the results by squaring each root. $$169$$
View solution