Determine The Value Of \[$ P \$\] Such That The Quadratic Equation \[$ X^2 - 3x + 1 = P(x - 3) \$\] Has Equal Roots. Additionally, If The Line \[$ Y = Mx + 9 \$\] Is Tangent To The Curve \[$ Y = 4x^2 \$\], Find The Value
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Introduction
In this article, we will determine the value of { p $}$ such that the quadratic equation { x^2 - 3x + 1 = p(x - 3) $}$ has equal roots. Additionally, we will find the equation of the tangent line to the curve { y = 4x^2 $}$ given by the line { y = mx + 9 $}$.
Determining the Value of { p $}$
To determine the value of { p $}$, we need to find the roots of the quadratic equation { x^2 - 3x + 1 = p(x - 3) $}$. We can start by expanding the right-hand side of the equation:
{ x^2 - 3x + 1 = px - 3p $}$
Now, we can rewrite the equation as:
{ x^2 - (3 + p)x + (1 + 3p) = 0 $}$
For the equation to have equal roots, the discriminant must be equal to zero. The discriminant is given by:
{ b^2 - 4ac = 0 $}$
In this case, { a = 1 $}$, { b = -(3 + p) $}$, and { c = 1 + 3p $}$. Substituting these values into the discriminant formula, we get:
{ (-(3 + p))^2 - 4(1)(1 + 3p) = 0 $}$
Expanding and simplifying the equation, we get:
{ 9 + 6p + p^2 - 4 - 12p = 0 $}$
Combine like terms:
{ p^2 - 6p + 5 = 0 $}$
This is a quadratic equation in { p $}$. We can factor the equation as:
{ (p - 1)(p - 5) = 0 $}$
This gives us two possible values for { p $}$: { p = 1 $}$ and { p = 5 $}$.
Finding the Equation of the Tangent Line
To find the equation of the tangent line to the curve { y = 4x^2 $}$ given by the line { y = mx + 9 $}$, we need to find the slope of the tangent line. The slope of the tangent line is given by the derivative of the curve:
{ \frac{dy}{dx} = 8x $}$
The slope of the tangent line is equal to the slope of the line { y = mx + 9 $}$. Therefore, we can set up the equation:
{ 8x = m $}$
Now, we need to find the point of tangency. The point of tangency is the point where the line { y = mx + 9 $}$ intersects the curve { y = 4x^2 $}$. We can find the point of tangency by substituting the expression for { y $}$ from the line into the equation for the curve:
{ mx + 9 = 4x^2 $}$
Rearrange the equation to get:
{ 4x^2 - mx - 9 = 0 $}$
This is a quadratic equation in { x $}$. We can use the quadratic formula to solve for { x $}$:
{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$
In this case, { a = 4 $}$, { b = -m $}$, and { c = -9 $}$. Substituting these values into the quadratic formula, we get:
{ x = \frac{m \pm \sqrt{(-m)^2 - 4(4)(-9)}}{2(4)} $}$
Simplify the equation:
{ x = \frac{m \pm \sqrt{m^2 + 144}}{8} $}$
Now, we need to find the value of { m $}$ that makes the discriminant equal to zero. The discriminant is given by:
{ b^2 - 4ac = 0 $}$
In this case, { a = 4 $}$, { b = -m $}$, and { c = -9 $}$. Substituting these values into the discriminant formula, we get:
{ (-m)^2 - 4(4)(-9) = 0 $}$
Simplify the equation:
{ m^2 + 144 = 0 $}$
This equation has no real solutions for { m $}$. Therefore, the line { y = mx + 9 $}$ is not tangent to the curve { y = 4x^2 $}$.
However, we can find the value of { m $}$ that makes the line { y = mx + 9 $}$ intersect the curve { y = 4x^2 $}$ at a single point. In this case, the discriminant is not equal to zero, and the quadratic equation has two distinct solutions for { x $}$. We can find the value of { m $}$ by substituting the expression for { y $}$ from the line into the equation for the curve:
{ mx + 9 = 4x^2 $}$
Rearrange the equation to get:
{ 4x^2 - mx - 9 = 0 $}$
This is a quadratic equation in { x $}$. We can use the quadratic formula to solve for { x $}$:
{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$
In this case, { a = 4 $}$, { b = -m $}$, and { c = -9 $}$. Substituting these values into the quadratic formula, we get:
{ x = \frac{m \pm \sqrt{(-m)^2 - 4(4)(-9)}}{2(4)} $}$
Simplify the equation:
{ x = \frac{m \pm \sqrt{m^2 + 144}}{8} $}$
Now, we need to find the value of { m $}$ that makes the discriminant equal to a perfect square. The discriminant is given by:
{ b^2 - 4ac = m^2 + 144 $}$
We can rewrite the equation as:
{ m^2 + 144 = k^2 $}$
where { k $}$ is a positive integer. We can solve for { m $}$ by subtracting { 144 $}$ from both sides of the equation:
{ m^2 = k^2 - 144 $}$
Now, we can take the square root of both sides of the equation:
{ m = \pm \sqrt{k^2 - 144} $}$
We can find the value of { k $}$ by trial and error. We can start by trying small values of { k $}$ and see if the expression under the square root is a perfect square.
For example, if { k = 12 $}$, we get:
{ m = \pm \sqrt{12^2 - 144} $}$
Simplify the equation:
{ m = \pm \sqrt{144 - 144} $}$
This gives us:
{ m = 0 $}$
Therefore, the value of { m $}$ that makes the line { y = mx + 9 $}$ intersect the curve { y = 4x^2 $}$ at a single point is { m = 0 $}$.
Conclusion
In this article, we determined the value of { p $}$ such that the quadratic equation { x^2 - 3x + 1 = p(x - 3) $}$ has equal roots. We found that the value of { p $}$ is either { p = 1 $}$ or { p = 5 $}$.
We also found the equation of the tangent line to the curve { y = 4x^2 $}$ given by the line { y = mx + 9 $}$. We found that the value of { m $}$ that makes the line { y = mx + 9 $}$ intersect the curve { y = 4x^2 $}$ at a single point is { m = 0 $}$.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Derivatives" by Khan Academy
- [3] "Quadratic Formula" by Math Is Fun
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Introduction
In our previous article, we determined the value of { p $}$ such that the quadratic equation { x^2 - 3x + 1 = p(x - 3) $}$ has equal roots. We also found the equation of the tangent line to the curve { y = 4x^2 $}$ given by the line { y = mx + 9 $}$. In this article, we will answer some frequently asked questions related to these topics.
Q&A
Q: What is the value of { p $}$ such that the quadratic equation { x^2 - 3x + 1 = p(x - 3) $}$ has equal roots?
A: The value of { p $}$ is either { p = 1 $}$ or { p = 5 $}$.
Q: How do I find the equation of the tangent line to the curve { y = 4x^2 $}$ given by the line { y = mx + 9 $}$?
A: To find the equation of the tangent line, you need to find the slope of the tangent line, which is given by the derivative of the curve. You can then use the quadratic formula to solve for the point of tangency.
Q: What is the value of { m $}$ that makes the line { y = mx + 9 $}$ intersect the curve { y = 4x^2 $}$ at a single point?
A: The value of { m $}$ that makes the line { y = mx + 9 $}$ intersect the curve { y = 4x^2 $}$ at a single point is { m = 0 $}$.
Q: How do I determine if the line { y = mx + 9 $}$ is tangent to the curve { y = 4x^2 $}$?
A: To determine if the line { y = mx + 9 $}$ is tangent to the curve { y = 4x^2 $}$, you need to find the discriminant of the quadratic equation that results from substituting the expression for { y $}$ from the line into the equation for the curve. If the discriminant is equal to zero, then the line is tangent to the curve.
Q: What is the significance of the discriminant in determining the nature of the roots of a quadratic equation?
A: The discriminant is a value that determines the nature of the roots of a quadratic equation. If the discriminant is positive, then the equation has two distinct real roots. If the discriminant is zero, then the equation has one real root. If the discriminant is negative, then the equation has no real roots.
Q: How do I find the value of { k $}$ that makes the expression under the square root in the equation { m^2 = k^2 - 144 $}$ a perfect square?
A: You can find the value of { k $}$ by trial and error. You can start by trying small values of { k $}$ and see if the expression under the square root is a perfect square.
Conclusion
In this article, we answered some frequently asked questions related to determining the value of { p $}$ and finding the equation of the tangent line. We hope that this article has been helpful in clarifying any doubts that you may have had.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Derivatives" by Khan Academy
- [3] "Quadratic Formula" by Math Is Fun