Problem 89
Question
Solve each equation by factoring. \(9 x^{2}+15 x+4=0\)
Step-by-Step Solution
Verified Answer
The solution to the equation \(9 x^{2}+15 x+4=0\) is \(x_1= -2/3\) and \(x_2= -2/3\)
1Step 1: Rewrite the Equation in Quadratic Form
Rewrite the quadratic equation \(9 x^{2}+15 x+4=0\) in standard form as \(ax^{2} + bx + c = 0\), where \(a=9\), \(b=15\), and \(c=4\).
2Step 2: Factorise the Quadratic Equation
The quadratic equation is factored as \((3x+2)(3x+2) = 0\). This was done by grouping the equation in pairs, factoring out the common terms, and then comparing the two results to arrive at the factorized form.
3Step 3: Solve the Factorised Equation
Having the equation in the form \((3x+2)(3x+2) = 0\), isolate each factor and equate it to zero to find the roots. \n This gives us two possible equations: \(3x+2 = 0\) and \(3x+2 = 0\). Solving for \(x\) in each equation gives the solution.\n Solve \(3x+2=0\), which results in \(x=-2/3\). Hence, the roots of the quadratic equation are \(x_1= -2/3\) and \(x_2= -2/3\).
Key Concepts
Quadratic EquationsFactoringSolutions of Quadratic Equations
Quadratic Equations
Quadratic equations are a type of polynomial equation, with the highest degree term being a square. This means the equation contains terms of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations form a parabola when graphed, and can have up to two solutions or roots.
Understanding quadratic equations is crucial, especially in algebra, as they form a foundation for higher-level math topics. Quadratics can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Each method provides different advantages depending on the specific equation and the information available.
In the given exercise, the quadratic equation \(9x^2 + 15x + 4 = 0\) is already in standard form. Recognizing the equation's structure helps in deciding which solving method to use, and ultimately, reveals the beauty of mathematical problem-solving.
Understanding quadratic equations is crucial, especially in algebra, as they form a foundation for higher-level math topics. Quadratics can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Each method provides different advantages depending on the specific equation and the information available.
In the given exercise, the quadratic equation \(9x^2 + 15x + 4 = 0\) is already in standard form. Recognizing the equation's structure helps in deciding which solving method to use, and ultimately, reveals the beauty of mathematical problem-solving.
Factoring
Factoring is one of the primary methods for solving quadratic equations. It involves rewriting a quadratic equation as a product of two binomial expressions. The initial step is to identify two numbers that multiply to \(a \times c\) and add up to \(b\) in the equation \(ax^2 + bx + c\).
The example equation \(9x^2 + 15x + 4 = 0\) is factored into \((3x+2)(3x+2) = 0\). This process requires finding common factors and grouping terms, aiming to simplify the quadratic into its component parts. Factoring is effective for equations where the above conditions align neatly, though not all quadratics are factorable using integers easily. In these cases, one might need to resort to other methods such as completing the square or using the quadratic formula.
The example equation \(9x^2 + 15x + 4 = 0\) is factored into \((3x+2)(3x+2) = 0\). This process requires finding common factors and grouping terms, aiming to simplify the quadratic into its component parts. Factoring is effective for equations where the above conditions align neatly, though not all quadratics are factorable using integers easily. In these cases, one might need to resort to other methods such as completing the square or using the quadratic formula.
Solutions of Quadratic Equations
Once a quadratic equation is factored, finding the solutions becomes straightforward. The factors obtained, in this case, \((3x+2)(3x+2) = 0\), indicate possible values for \(x\).
To find the solutions, set each factor of the equation to zero separately. For the two factors here, this means solving \(3x+2 = 0\) to get the solution \(x = -\frac{2}{3}\). Since both factors are the same, it means we have a repeated root, confirming the solution \(x_1 = -\frac{2}{3}\) and \(x_2 = -\frac{2}{3}\).
This demonstrates a crucial aspect of quadratic solutions: a quadratic can have two distinct solutions, one repeated solution, or no real solution depending on the discriminant. In this specific problem, the repeated solution highlights a situation where the parabola touches the x-axis at a single point. Understanding this helps in visualizing and solving more complex quadratics efficiently.
To find the solutions, set each factor of the equation to zero separately. For the two factors here, this means solving \(3x+2 = 0\) to get the solution \(x = -\frac{2}{3}\). Since both factors are the same, it means we have a repeated root, confirming the solution \(x_1 = -\frac{2}{3}\) and \(x_2 = -\frac{2}{3}\).
This demonstrates a crucial aspect of quadratic solutions: a quadratic can have two distinct solutions, one repeated solution, or no real solution depending on the discriminant. In this specific problem, the repeated solution highlights a situation where the parabola touches the x-axis at a single point. Understanding this helps in visualizing and solving more complex quadratics efficiently.
Other exercises in this chapter
Problem 89
List all possible rational roots for each equation. Then use the Rational Root Theorem to find each root. $$ 3 x^{3}-5 x^{2}-4 x+4=0 $$
View solution Problem 89
Let \(g(x)=x-3\) and \(h(x)=x^{2}+6 .\) Find \((h \circ g)(1)\) $$\begin{array}{llll}{\text { A. }-14} & {\text { B. } 4} & {\text { C. } 5} & {\text { D. } 10}
View solution Problem 89
Which expression is NOT equivalent to \(\sqrt[6]{81 x^{4} y^{8}} ?\) \(\begin{array}{llll}{\text { E. }\left(3 x y^{2}\right)^{\frac{2}{3}}} & {\text { G. }(3 x
View solution Problem 90
Solve using the Quadratic Formula. \(x^{2}+10 x+11=0\)
View solution