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Chapter 17: Problem 478

Solve \(x^{2}-7 x+10=0\)

Short Answer

Expert verified
The solutions to the given quadratic equation \(x^2 - 7x + 10 = 0\) are \(x = -2\) and \(x = -5\).

Step by step solution

01

Identify the equation format

Identify that the given equation is a quadratic equation in the format of \(ax^2 + bx + c = 0\). In our case, \(a = 1\), \(b = -7\), and \(c = 10\).
02

Factor the quadratic equation

Now, we need to factor the quadratic equation into the form \((x - m)(x - n) = 0\). To find the two factors m and n, look for two numbers whose product is equal to "a * c" and whose sum is equal to "b". In our case, the product is \(1 * 10 = 10\) and the sum is \(-7\). The two numbers are \(-2\) and \(-5\), since \(-2 * -5 = 10\) and \(-2 + (-5) = -7\). Now, rewrite the quadratic equation as the product of these factors: \((x - (-2))(x - (-5))\), which simplifies to \((x + 2)(x + 5) = 0\).
03

Identify solutions

The equation \((x + 2)(x + 5) = 0\) is satisfied when either one of its factors is equal to zero. That is, we need to solve the equations \(x + 2 = 0\) and \(x + 5 = 0\). From the first equation, \(x + 2 = 0\), we get \(x = -2\). From the second equation, \(x + 5 = 0\), we get \(x = -5\).
04

Write the final solution

The solutions to the given quadratic equation \(x^2 - 7x + 10 = 0\) are \(x = -2\) and \(x = -5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Quadratics
Quadratic equations often come in the form of \(ax^2 + bx + c = 0\), which might seem intimidating at first. A super helpful way to solve them is by factoring. Factoring involves rewriting the quadratic as a product of two binomials. The trick is to find two numbers that multiply together to give you \(a \cdot c\), and add up to \(b\).
In our specific equation \(x^2 - 7x + 10 = 0\), identifying these two numbers is key. Here, you look for numbers that multiply to 10 (\(a \cdot c\)) and add up to -7 (\(b\)).
  • Two numbers that work are -2 and -5.
By factoring the expression as \((x - 2)(x - 5) = 0\), we鈥檝e turned a quadratic equation into a form that鈥檚 much simpler to solve.
Solving Equations
Solving quadratic equations often feels like unraveling a mystery. Once you鈥檝e factored the quadratic equation into \((x - 2)(x - 5) = 0\), you can solve it using the Zero Product Property. The principle here is simple: if the product of two numbers is zero, at least one of the numbers must be zero.
  • This means you set each factor equal to zero: \(x - 2 = 0\) or \(x - 5 = 0\).
  • Resolving these simple equations gives you the solutions \(x = 2\) and \(x = 5\).
This systematic approach helps find the exact values where the quadratic equation equals zero.
Roots of Quadratic
The solutions we find 鈥� in this case, \(x = 2\) and \(x = 5\) 鈥� are called the roots of the quadratic equation. These roots tell us where the parabola represented by \(x^2 - 7x + 10 = 0\) intersects the x-axis.
  • A quadratic equation can have two, one, or no real roots.
  • In this specific instance, there are two distinct real roots.
Recognizing the roots provides not only solutions but also insights into the graph's behavior of the equation.
Algebraic Expressions
Algebraic expressions form the backbone of many math problems, especially quadratics. They consist of variables and coefficients organized in a logical manner. In quadratic equations like \(x^2 - 7x + 10\), understanding how each part interacts is important.
  • The term \(x^2\) is called the quadratic term and it shapes the parabola.
  • The term \(-7x\) is the linear term and affects the symmetry.
  • The constant 10 shifts the parabola vertically on the graph.
Together, these components allow us to manipulate and solve the equation efficiently through various methods, like factoring and solving.

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Most popular questions from this chapter

Solve \(\left(1-\mathrm{a}^{2}\right)(\mathrm{x}+\mathrm{a})-2 \mathrm{a}\left(1-\mathrm{x}^{2}\right)=0\)

Find a quadratic equation whose roots are \(3+2 \sqrt{3}\) and \(3-2 \sqrt{3}\)

Without solving the equation \(2 x^{2}-3 x+5=0\) determine the nature of its roots.

Find the sum and the product of the roots in each of the following equations: \(x^{2}-3 x+2=0,2 x^{2}+8 x-5=0\), and \(\sqrt{2} \mathrm{x}^{2}+5 \mathrm{x}-\sqrt{8}=0\)

Solve the equation \(\mathrm{x}^{2}+5 \mathrm{x}+6=0\) by the quadratic formula.
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