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.

Apply the method of completing the square to solve quadratic equations by transforming them into a square form and taking square roots, as shown with x^2-3x+2=0.

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.

Learn the quadratic formula method for solving ax^2+bx+c=0 by completing the square, deriving x equals (-b ± sqrt(b^2-4ac)) divided by 2a, and applying to x^2-3x+2, yielding roots 2 and 1.

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 solve 2x^2 + 5√3 x + 6 = 0, yielding the roots -√3/2 and -2√3.

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.

Compute the discriminant of quadratic equations using standard form and d = b^2 - 4ac. Practice with four example equations to identify a, b, c and confirm results.

Compute the discriminant for a quadratic equation in standard form. With a=4, b=-5, c=3, the discriminant is -23, so the roots are imaginary.

Show that the equation 2x^2 -6x +3 = 0 has real roots by evaluating the discriminant, then find the roots x = (3 ± sqrt(3))/2.

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.

Transform the equation 1/(x+a) + 1/(x+b) = 1/c into standard quadratic form. Prove the product of the roots equals minus 1/(2a^2 + b^2) using the sum of the roots.

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.

Form a quadratic from given roots by applying x^2 minus (sum of roots) x plus (product of roots); example with -4 and 6 yields x^2 - 2x - 24.

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.

Derive that alpha and beta have sum 5 and product 3, then form the equation with roots alpha/beta and beta/alpha: 3x^2 - 19x + 3 = 0.

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.

Solve a problem where natural numbers differ by five and the difference of their reciprocals is 1/10; set them as x and x−5, derive a quadratic, and find 5 and 10.

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.

Solve a real-world quadratic problem by equating times for two trains; set 192/(x-20) - 192/x = 48/60, and find the Deccan Queen's speed as 80 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.

This example shows that for the quadratic equation ax^2+bx+c=0 with roots alpha and beta, the expression a(alpha^2/beta+beta^2/alpha)+b(alpha/beta+beta/alpha) equals b, using alpha+beta=-b/a and alpha beta=c/a.

Learn the conditions for common roots in two quadratic equations, including one or both shared roots, using cross multiplication and coefficients a1, b1, c1 and a2, b2, c2.

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.

Solve quadratic radical equations by setting y = 2x^2 + 5x, square both sides, and confirm the roots x = 2 and x = -9/2.

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.

Investigate quadratic equations through an example that determines the number of roots by factoring and splitting the middle term, concluding that the equation has zero roots.

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.

Solve the equation x^{log_x((1 - x)^2)} = 9 by applying base-x logarithms to get (1 - x)^2 = 9, then solve the quadratic to find x = -2 or x = 4.

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

Identify a common root alpha for the equations x^2+ax+b=0 and x^2+bx+a=0, then apply cross-multiplication to express alpha from both equations. Equating these expressions yields a+b = -1.

Apply the quadratic formula to x^2 + 2√3 x + 3 = 0, compute b^2 - 4ac = 12 - 12, and obtain a repeated irrational root -√3.

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.

Solve the system x^2 + y^2 = 25 and xy = 12 by substitution, yielding x^2 = 16 or 9, so x = ±4 or ±3 with corresponding y values.

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.