Use the quadratic formula \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to find the values of y. Plug in the coefficients:\[y = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(4)(-3)}}{2(4)}\]Simplify inside the square root first: \[y = \frac{8 \pm \sqrt{64 + 48}}{8}\]\[y = \frac{8 \pm \sqrt{112}}{8}\]
04
Simplify the Square Root
Simplify \(\sqrt{112}\) as follows: \[\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}\]. Plugging this back, we get: \[y = \frac{8 \pm 4\sqrt{7}}{8}\].
05
Simplify the Solutions
Simplify the fractions: \[y = 1 \pm \frac{\sqrt{7}}{2}\]Therefore, the solutions are: \[y = 1 + \frac{\sqrt{7}}{2}\] and \[y = 1 - \frac{\sqrt{7}}{2}\]
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. Quadratic equations are in the form \[ ax^2 + bx + c = 0\]. The quadratic formula to find the solutions is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
\( a \) is the coefficient of \( x^2 \)
\( b \) is the coefficient of \( x \)
\( c \) is the constant term
The symbol \( \pm \) means we need to calculate two values: one with addition and one with subtraction. These two values are the solutions to the quadratic equation.
Standard Form
The standard form of a quadratic equation is \[ ax^2 + bx + c = 0\]. To solve any quadratic equation using the quadratic formula, you first need to rewrite it in this form. For example, given the equation \[ 4y^2 - 8y = 3\]:1. Subtract 3 from both sides to get:\[ 4y^2 - 8y - 3 = 0\]Now the equation is in standard form where \( a = 4 \), \( b = -8 \), and \( c = -3 \). This step is crucial as it prepares the equation for the application of the quadratic formula.
Simplifying Square Roots
When working with quadratic equations, you will often need to simplify square roots. This process involves finding the prime factors of the number inside the square root and simplifying. For example, simplify \( \sqrt{112} \):
Factorize 112 as \( 112 = 16 \cdot 7 \)
Take the square root of each factor:\( \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} \)
Simplify further as \( \sqrt{16} = 4 \), so the result is \( 4\sqrt{7} \)
This simplified form can then be used in further calculations, like when completing the quadratic formula.
Identifying Coefficients
Before using the quadratic formula, it's essential to correctly identify the coefficients of the quadratic equation in standard form. The standard form is \[ ax^2 + bx + c = 0\], and you need to find the values of \( a \), \( b \), and \( c \).For instance, in the equation \[ 4y^2 - 8y - 3 = 0 \]:
The coefficient \( a \) (of \( y^2 \)) is 4
The coefficient \( b \) (of \( y \)) is -8
The constant term \( c \) is -3
Accurately identifying these values allows you to correctly substitute them into the quadratic formula and solve the equation correctly.
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Most popular questions from this chapter
College Value The difference, \(d\), in median earnings, in \(\$ 1000\) s, betweenhigh school graduates and college graduates can be approximated by \(d=-0.012x^{2}+0.828 x+15.750,\) where \(x\) is the number of years after \(1980 .\) Basedon this model, estimate to the nearest year when the difference in medianearnings was \(\$ 25,000\).Discuss the step in the solving process for radical equations that leads tothe possibility of extraneous solutions. Why is there no such possibility forlinear and quadratic equations?Solve: \(\sqrt{3 x+5}-\sqrt{x-2}=\sqrt{x+3}\)Find the real solutions, if any, of each equation. $$ 3 z^{3}-12 z=-5 z^{2}+20 $$Constructing a Border around a Garden A landscaper, who just completed arectangular flower garden measuring 6 feet by 10 feet, orders 1 cubic yard ofpremixed cement, all of which is to be used to create a border of uniformwidth around the garden. If the border is to have a depth of 3 inches, howwide will the border be? (Hint: 1 cubic yard \(=27\) cubic feet)
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If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions..
If an inequality has no real solution, this means that there are no numbers that can be substituted into the inequality to make the statement true. If an inequality has all real numbers as the solution, this means that every real number can be substituted into the inequality to make a true statement.
The value of the discriminant shows how many roots f(x) has: - If b2 – 4ac > 0 then the quadratic function has two distinct real roots. - If b2 – 4ac = 0 then the quadratic function has one repeated real root. - If b2 – 4ac < 0 then the quadratic function has no real roots.
There is no real number whose square is negative. Therefore for this equation, there are no real number solutions. Hence, the expression (b2 – 4ac) is called the discriminant of the quadratic equation ax2 + bx + c = 0. Its value determines the nature of roots as we shall see.
This type of equation is never true, no matter what we replace the variable with. As an example, consider 3x + 5 = 3x - 5. This equation has no solution. There is no value that will ever satisfy this type of equation.
For example, in an aqueous sugar solution, the sugar molecules are evenly distributed among the water molecules such that each volume part of the solution contains an equal number of sugar or water molecules.
In mathematics, a real solution or real root refers to the solution of an equation that is a real number. A real number is a number that can be plotted on a number line. All numbers belong to the set of numbers known as the real number system.
Real numbers can be defined as the union of both rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”. All the natural numbers, decimals and fractions come under this category. See the figure, given below, which shows the classification of real numerals.
Some equations have no solutions. In these equations, there is no value for the variable that makes the equation true. You can tell that an equation has no solutions if you try to solve the equation and get a false statement. Let's try it!
The coefficients are the numbers alongside the variables. The constants are the numbers alone with no variables. If the coefficients are the same on both sides then the sides will not equal, therefore no solutions will occur.
If the value of the discriminant is positive, there are two real solutions for x, meaning the graph of the solution has two distinct x-intercepts. If the value of the discriminant is zero, there is one real solution for x, meaning the graph of the solution has one x-intercept.
If you are asked if a point is a solution to an equation, we replace the variables with the given values and see if the 2 sides of the equation are equal (so is a solution), or not equal (so not a solution). A solution to a system of equations means the point must work in both equations in the system.
If an equation has real solutions it just means that its graph crosses the x axis at some point.If it doesn't have real solutions, the graph of the equation never crosses the x axis.
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