Quadratic Equation -A complete course on Quadratic Equations

Factor the quadratic into x+2 and 3x-5, with their product equal to zero. Solve each factor to obtain the roots x = -2 and x = 5/3.

This lecture shows how to convert a quadratic to standard form, split the middle term, and factor to find the roots x = 5/2 or x = 8/3.

Factorize the quadratic by splitting the middle term, group to reveal a repeated factor, and find equal roots at x = 3/2.

Cross-multiply the equation to form a quadratic, simplify to x^2-4x-5=0, factor to (x-5)(x+1), and find x=5 or x=-1.

Solving x^2 -10x -2 = 0 by completing the square forms the perfect square (x-5)^2 = 27 and yields the roots 5 ± 3√3.

Learn to solve 9x^2 - 15x + 6 = 0 by completing the square, dividing by nine to set x^2 coefficient to one, obtaining roots 1 and 2/3.

Using completing the square on 4x^2+3x+5=0, divide by four, form (x+3/8)^2, and show the right-hand side is negative, hence the equation has no real roots.

Solve a quadratic equation by applying the quadratic formula to the standard form ax^2+bx+c=0, identify a, b, c, and compute the roots 2/9 and -1.

Apply the quadratic formula to x^2 + 6x + 6 = 0 and obtain the roots -3 ± sqrt(3) after simplifying the discriminant.

Apply the quadratic formula to a = p^2, b = -p^2 - q^2, c = -q^2. Compute the discriminant D = (p^2 + q^2)^2 and find roots x = q^2/p^2 and x = -1.

Clear denominators with the LCM to turn the equation into 2x^2 -15x +25 = 0, then apply the quadratic formula to obtain roots 5 and 5/2, with x ≠ 2,4.

Learn how equal roots occur when the discriminant is zero, by comparing to the standard quadratic ax^2+bx+c=0. Derive that ad=bc, implying a/b=c/d.

In example 9, determine the discriminant to show that the quadratic in x has no real roots when a ≠ b, highlighting the condition for quadratic equations.

Analyze the discriminant of ax^2+bx+c=0 to identify real and equal roots, and prove that either a is zero or a^3+b^3+c^3 equals 3abc.

Show that if two quadratics have real roots simultaneously, then b^2 equals ac by comparing discriminants: d1 ≥ 0 yields b^2 ≥ ac, d2 ≥ 0 yields b^2 ≤ ac.

The lecture derives that for the quadratic (1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0, equal roots require the discriminant to be zero, leading to c^2 = a^2(1+m^2).

Determine the values of m for which the roots are equal by setting the discriminant to zero after bringing the equation to standard form, which yields 3 or 5.

Use alpha plus beta equals minus b over a and alpha beta equals c over a. Compute the sum as 4/3 and the product as 3 for a=3, b=-4, c=9.

Examine the quadratic l x^2 + n x + n = 0 with roots alpha and beta, given alpha/beta = p/q, and prove sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.

Show that for the quadratic 4x^2+2x-1=0, if alpha is a root then the other root equals alpha^3 minus 3 alpha, using root-sum relations.

learn how to form a quadratic equation from given roots alpha and beta, using x^2 - (alpha+beta)x + alpha beta = 0, with examples and conjugate root rules.

From roots three plus under root two and three minus under root two, form a quadratic using their sum six and product seven: x^2 - 6x + 7 = 0.

Form the quadratic with complex conjugate roots 3-2i and 3+2i by using their sum 6 and product 13, yielding x^2-6x+13=0.

Use the sum and product of roots to derive the quadratic whose roots are alpha^2+2 and beta^2+2 for 2x^2-3x-6=0, avoiding explicit root calculation.

Demonstrates that the quadratic with roots q/(p - alpha) and q/(p - beta) has sum p and product q, yielding x^2 - p x + q = 0.

Solve for two consecutive even numbers whose squares sum to 340. Let the numbers be x and x+2; set x^2+(x+2)^2=340 and obtain 12 and 14.

Set sister's age as x and girl's as 2x. Four years later the product equals 160, leading to x = 6, so sister is 6 and the girl is 12.

Solve a quadratic age problem: with the sum of ages 45 and the product five years ago equal to 124, the man is 36 and the son is 9.

Solve the quadratic L^2 - 41L + 400 = 0 to find the rectangular field’s dimensions: length 25 m and breadth 16 m.

Solve for the usual speed given 3 km in less time when speed is 1 km/h faster; use distance equals speed times time and solve x^2+x-12=0, yielding x=3 km/h.

Explore symmetric expressions of quadratic roots, swapping alpha and beta leaves expressions unchanged, and learn to express alpha^2+beta^2, alpha^3+beta^3, and related terms using alpha+beta and alpha beta.

Find expressions in terms of a, b, and c for the roots alpha and beta of ax^2+bx+c=0. Show 1/alpha+1/beta = -b/c and 1/alpha^2+1/beta^2 = (b^2-2ac)/c^2.

Derive expressions in terms of a, b, c from a quadratic with roots alpha and beta, using sum and product relations to obtain first expression b/(ac) and second -2/a.

Derive alpha plus beta and alpha beta from the standard form of a quadratic with roots alpha and beta, then prove (alpha+1)(beta+1) equals 1−c and the related expression equals one.

Determine the values of m that ensure the two quadratic equations share a common root; equate expressions for the common root to obtain m = ±1/√2.

Solve a radical equation reducible to a quadratic by squaring both sides, forming x^2 -6x +5 = 0 to yield x = 5 or 1, with verification.

This example shows solving a radical equation by squaring both sides, simplifying to a quadratic, and verifying roots to discard extraneous solutions, yielding x=4 as the valid root.

Solve equation five plus two root six to power x^2 minus three plus five minus two root six to power x^2 minus three equals ten; yields x = ±2 or ±√2.

Let y = 5^x to convert the equation into 5y^2 - 26y + 25 = 0, solve for y to get 25 or 1/5, then x = 2 or -1.

Rationalize a complex square-root expression, simplify to x^2 = a^2, and conclude x equals plus or minus a (with x not equal to zero).

Derive the equation with roots (alpha-1)/(alpha+1) and (beta-1)/(beta+1) for the given quadratic x^2-2x+3=0 by computing their sum and product, yielding 3x^2-2x+1=0.

Establish x as the infinite expression 3 plus 1 over x, derive x^2 - 3x - 1 = 0, and obtain positive root x = (3 + sqrt(13))/2.

Determine the minimum value of the quadratic 2x^2 - 3x + 1 for real x by using the discriminant, yielding y = -1/8 as the minimum.

Analyze a rational equation by canceling the common factor (x-1) and applying the domain restriction x ≠ 1, then conclude there is no solution.

Solve the infinite nested radical by setting x = sqrt(2 + x) and squaring to get x^2 = 2 + x, leading to x = 2 after rejecting minus one.

Substitute y = x^(1/3) to convert the equation into a quadratic, solve for y, then obtain x = -8 or 1.

Use the quadratic formula with a = p - q, b = q - r, c = r - p; roots are (r - p)/(p - q) and 1.

Analyze when a quadratic in x has two rational factors by using the discriminant and a perfect-square form, and find m values of -2 and 6.

Factor the quadratic in |x|: |x|^2−3|x|+2=0 into (|x|−2)(|x|−1)=0, giving |x|=2 or |x|=1, hence x=±2 or ±1, four real solutions.

Transform the equation x + 1/x = 2 into x^2 - 2x + 1 = 0, recognize a perfect square, and find the double root x = 1.

Explore how the discriminant of a quadratic equation determines roots, showing that with odd integers a and b, D becomes a perfect square and yields rational roots.