Problem 105
Question
Will help you prepare for the material covered in the next section. Solve by completing the square: \(y^{2}-6 y-4=0\)
Step-by-Step Solution
Verified Answer
The solutions to the equation \(y^{2}-6 y-4=0\) are \(y = 3 + \sqrt{13}\) and \(y = 3 - \sqrt{13}\)
1Step 1: Rearrange the Equation
In order to complete the square, rearrange the equation to the form \(y^2 - By = C\), which becomes \(y^2 - 6y = 4\)
2Step 2: Complete the Square
To complete the square on the left side of the equation, add \((B/2)^2\) to both sides of the equation. Here, \(B = -6\), so \((B/2)^2 = (6/2)^2 = 9\). Therefore, the equation will become \(y^2 - 6y + 9 = 4 + 9\), which simplifies to \((y-3)^2 = 13\)
3Step 3: Solve for y
Take the square root of both sides of the equation to solve for y: \(y - 3 = \pm\sqrt{13}\). This means that \(y = 3 \pm\sqrt{13}\)
Key Concepts
Quadratic EquationsFactoringSquare Roots
Quadratic Equations
Quadratic equations form a fundamental part of algebra and are written in the standard form as \(ax^2 + bx + c = 0\). They involve a squared term (\(x^2\)), which distinguishes them from linear equations. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, and applying the quadratic formula.
Each quadratic equation has a parabolic graph, which can open upwards or downwards depending on the sign of the \(a\) coefficient. When working with quadratic equations, the roots are the \(x\)-intercepts of this parabolic graph. These are the values that satisfy the equation \(ax^2 + bx + c = 0\). Understanding how to manipulate these equations to find the roots is crucial.
Each quadratic equation has a parabolic graph, which can open upwards or downwards depending on the sign of the \(a\) coefficient. When working with quadratic equations, the roots are the \(x\)-intercepts of this parabolic graph. These are the values that satisfy the equation \(ax^2 + bx + c = 0\). Understanding how to manipulate these equations to find the roots is crucial.
- Standard Form: \(ax^2 + bx + c = 0\)
- Solutions (Roots): points where \(y = 0\)
- Methods: factoring, completing the square, quadratic formula
Factoring
Factoring is an essential algebraic skill used to express a quadratic equation in terms of its factors, which can sometimes be more straightforward than solving the equation directly.
For example, if a quadratic can be factored into the form \((x - r)(x - s) = 0\), then it's easy to solve for \(x\) by setting each factor to zero: \(x = r\) or \(x = s\). This method finds the roots by identifying two binomials that multiply to give the original equation.
However, not all quadratics are easily factorable, which is where methods like completing the square or using the quadratic formula come into play. Factoring requires recognizing patterns or using techniques such as the method of grouping, and is often the fastest way to solve simple quadratics.
For example, if a quadratic can be factored into the form \((x - r)(x - s) = 0\), then it's easy to solve for \(x\) by setting each factor to zero: \(x = r\) or \(x = s\). This method finds the roots by identifying two binomials that multiply to give the original equation.
However, not all quadratics are easily factorable, which is where methods like completing the square or using the quadratic formula come into play. Factoring requires recognizing patterns or using techniques such as the method of grouping, and is often the fastest way to solve simple quadratics.
- Easy for simple quadratics
- Expresses equation as a product of two binomials
- May not be applicable for all quadratic equations
Square Roots
Square roots play a critical role in solving quadratic equations, especially when using the method of completing the square.
Once an equation is rewritten in the form of a perfect square trinomial like \((y-3)^2 = 13\), finding \(y\) involves taking the square root of both sides. This yields two solutions: one positive and one negative, because every positive number has two square roots of opposite signs.
The step of taking the square root simplifies the process of isolating the variable. It is key to note that when considering square roots, both the principal (positive) root and the negative counterpart must be taken into account to capture all possible solutions.
Once an equation is rewritten in the form of a perfect square trinomial like \((y-3)^2 = 13\), finding \(y\) involves taking the square root of both sides. This yields two solutions: one positive and one negative, because every positive number has two square roots of opposite signs.
The step of taking the square root simplifies the process of isolating the variable. It is key to note that when considering square roots, both the principal (positive) root and the negative counterpart must be taken into account to capture all possible solutions.
- Essential for completing the square
- Produces two solutions: positive & negative
- Helps isolate variables in quadratic equations
Other exercises in this chapter
Problem 104
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